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Author(s): Puska, M. J. & Seitsonen, Ari P. & Nieminen, Risto M.

Title: Electron-positron Car-Parrinello methods: Self-consistent treatment ofcharge densities and ionic relaxations

Year: 1995

Version: Final published version

Please cite the original version:Puska, M. J. & Seitsonen, Ari P. & Nieminen, Risto M. 1995. Electron-positronCar-Parrinello methods: Self-consistent treatment of charge densities and ionicrelaxations. Physical Review B. Volume 52, Issue 15. 10947-10961. ISSN 1550-235X(electronic). DOI: 10.1103/physrevb.52.10947.

Rights: © 1995 American Physical Society (APS). This is the accepted version of the following article: Puska, M. J. &Seitsonen, Ari P. & Nieminen, Risto M. 1995. Electron-positron Car-Parrinello methods: Self-consistenttreatment of charge densities and ionic relaxations. Physical Review B. Volume 52, Issue 15. 10947-10961.ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.52.10947, which has been published in final form athttp://journals.aps.org/prb/abstract/10.1103/PhysRevB.52.10947.

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PHYSICAL REVIEW 8 VOLUME 52, NUMBER 15 15 OCTOBER 1995-I

Electron-positron Car-Parrinello methods: Self-consistent treatmentof charge densities and ionic relaxations

M. J. PuskaLaboratory ofPhysics, Helsinki University of Technology, FIX 0215-0 Espoo, Finland

Ari P. SeitsonenFritz Habe-r Insti-tut der Max Plane-k Gesel-lschaft, Faradayweg 4-6, D 14 19-5 Berlin Dah-lem, Germany

and Laboratory ofPhysics, Helsinki University of Technology, FIX 02150-Espoo, Finland

R. M. NieminenLaboratory ofPhysics, Helsinki University of Technology, FIN 02150-Espoo, Finland

(Received 22 May 1995)

A calculation method based on the two-component density-functional theory is presented for electronsystems with a localized positron. Electron-ion and positron-ion interactions are described by norm-conserving pseudopotentials and full ionic potentials, respectively. The electron and positron densitiesare solved self-consistently using a plane-wave expansion for electron and a real-space grid method forpositron wave functions, respectively. The forces on ions exerted by a positron trapped at an open-volume defect and the ensuing ionic relaxations are determined using first-principles methods. In thecase of semiconductors, the self-consistent solution of electron and positron densities as well as the ionicpositions are found to depend strongly on the treatment of the electron-positron correlation in construct-ing the effective potentials. We consider several approximations to the correlation energy while demon-strating the method by calculations for a positron trapped by a Ga vacancy in CxaAs. Especially, theeffects on the observable positron annihilation characteristics, i.e., positron lifetimes, core annihilationline shapes, and two-dimensional angular correlation maps are discussed.

I. INTRODUCTION

The experimental methods based on positron annihila-tion are powerful tools in investigating the electronicstructures of solids, especially the properties of vacancy-type defects in materials. ' The vacancy-type defectstrap positrons due to the reduction of the repulsivenucleus-positron interaction. The trapping leads to an in-crease of the positron lifetime because the average elec-tron density at a vacancy-type defect is lower than in theperfect crystal. At the same time the ratio between theannihilations with core and valence electron changes.The momentum density of the valence electrons sampledby the localized positron also deviates from its bulkvalue, and is reAected in the angular correlation of an-nihilation radiation (ACAR). As the information on elec-tronic properties is indirect, comparisons with theoreticalpredictions for the positron annihilation characteristicsare needed for proper interpretation of the experimentalfindings.

The quantitative modeling of vacancy-type defectswith a trapped positron is a complicated many-bodyproblem as it involves the simultaneous determination ofthe electronic structure and the ionic positions aroundthe defect. This complexity is especially true for defectsin semiconductors that show (already without thetrapped positron) a richness of structures and phenomenarelated to the coupling between the electronic states andionic degrees of freedom. On top of this, the introduc-

tion of the positron into a defect may a6ect considerablyits electronic and ionic structure. The calculations forpositron states at defects in semiconductors have thus farbeen mainly model calculations, in which, e.g. , theatomic positions are given as an input to the calculation,instead of full first-principles calculations without anypreassumptions.

The principles for calculating positron states and an-nihilation characteristics have been laid down as thetwo-component density-functional formalism. Althoughthe formalism has been presented already several yearsago, its applications have been quite scarce thus far. Re-cently, Gilgien et QI. ' demonstrated the strength of themethod in the case of the singly negative As vacancy inGaAs. The calculation included also the self-consistentrelaxation of the atoms around the vacancy, so that itwas the first ab initio calculation of its kind. In semicon-ductors the relaxation of atoms plays a crucial role indetermining the properties of defects and the importanceof the positron-induced relaxation, which was earlier dis-cussed serniempirically, " was treated by Gilgien et al.from first principles.

In the two-component density-functional theory thebasic quantities are the positron and electron densities.In the case of a localized positron at a lattice defect, thepositron density is finite near the defect and has an e6'ecton the average electron density. The electron and posi-tron densities have to be determined simultaneously, re-quiring the self-consistency of the solution (in the case ofdelocalized states the positron density is vanishingly

0163-1829/95/52(15)/10947(15)/$06. 00 52 10 947 1995 The American Physical Society

M. J. PUSKA, ARI P. SEITSONEN, AND R. M. NIEMINEN 52

small everywhere, and does not affect the bulk electronicstructure). The self-consistency includes the CoulombicHartree interactions and the different exchange andcorrelation interactions among the charge densities. As amatter of fact, it turned out during this work that eventhe qualitative results can depend sensitively on the treat-ment of the electron-positron correlation energy. Therelative importance of the electron-positron correlationenergy is much larger than the electron-electron correla-

. tion energy. This is natural, because the magnitude ofthe former, calculated per positron in a homogeneouselectron gas, is an order of magnitude larger than theelectron-electron correlation energy per electron for thesame electron gas density.

We have used for the electron-positron correlation en-ergy the local-density-approximation (LDA) scheme sug-gested by Boronski and Nieminen (BN), in which thecorrelation energy functional and the corresponding con-tributions to the effective electron and positron potentialsdepend on the electron and positron densities. For thisscheme we introduce a two-dimensional interpolationform for the correlation energy based on Lantto's hyper-netted chain approximation calculations' for homogene-ous electron-positron plasmas with varying positron andelectron densities. We have also carried out calculationsusing the model suggested by Gilgien et al. ' (GGGC)for the electron-positron correlation energy. In this mod-el the positron potential does not depend explicitly on thepositron density, and in the case of localized positronsthe effective potential is much more attractive than in theBN scheme. In model calculations, in which the ions arenot allowed to relax due to positron trapping it is cus-tomary to apply the "conventional" scheme (the CONVscheme) in which the electron density is not at all afFectedby the positron. Then the positron potential dependsonly on the electron density, unperturbed by the presenceof the positron. The results of this approach are in factvery similar to those obtained by the BN scheme (seebelow). Finally, we have made calculations also by com-pletely omitting the electron-positron correlation in con-structing the eff'ective potentials (the NOCORR scheme).This leads to results that are in many aspects close tothose obtained by the GGGC scheme.

The enhancement of the electron density at the posi-tron, or the contact value of the electron-positron pair-correlation function, is another important electron-positron correlation quantity. It is needed in calculatingthe positron annihilation characteristics. In the BNscheme the enhancement depends on the electron andposltroil densities, and lt is calculated by using an LDAinterpolation form based on Lantto's hypernetted-chain-approximation calculations. According to Lantto's re-sults the contact density decreases remarkably when thepositron density, at a given constant electron density, in-creases. In the self-consistent calculations, the increaseof the contact density due to the rise of the average elec-tron density at a localized positron is effectively compen-sated by the decrease of the enhancement factor so thatthe positron lifetime obtained is close to that obtained bythe CONV method. In the GGGC scheme the enhance-ment is calculated assuming that the positron density is

vanishingly small. This leads to double counting in thesense that the contact density at a localized positron in-creases due to the increase of the average electron densityand at the same time the maximum strength for theenhancement is used. This leads to overestimation in theannihilation rate when compared with the BN scheme.However, this overestimation compensates the large de-crease of the annihilation rate due to the stronger posi-tron localization in the GGGC scheme. Thus, there is afeedback effect due to the fact that the positron densityrelaxes following the electron density.

Due to the compensating and feedback effects the posi-tron lifetime is not very sensitive to the model used forthe electron-positron correlation if the ionic positions arekept fixed. On the contrary, the intensity of core annihi-lations relative to the total annihilation rate dependsstrongly on the degree of the localization of the positronwave function and thereby on the correlation model used.The two-dimensional ACAR (2D-ACAR) plots for theannihilations with valence electrons also reAect the locali-zation of the positron wave function.

This work presents methods for the self-consistent cal-culations of the electron and positron states and the ionicstructures in the case of vacancy-type defects insolids. The modern first-principles molecular-dynamicsmethods' ' applied within the frame of the two-component density-functional formalism are well suitedfor the actual solution of the system. Moreover, in thiswork we carefully discuss the above questions related tothe electron-positron correlation, i.e., the buildup of thepotential trapping the positron and the enhancement ofthe electron density at the positron determining the posi-tron annihilation characteristics. The discussion is basedon the results for a triply negative Ga vacancy in GaAs,which is an important defect studied extensively by posi-tron annihilation measurements' ' and which due to itssimple breathing-mode-only type relaxation pattern iswell suited also for model studies. We will quantify ourdiscussion and make comparisons with experiments bycalculating the positron lifetimes, the core annihilationline shapes, and the 2D-ACAR maps with different mod-els for electron-positron correlation.

The organization of the present paper is as follows: InSec. II the computational methods are presented. It in-cludes a review of the two-component density-functionaltheory, presentation of the new interpolation forms forthe electron-positron correlation, discussion of thedifferent schemes related to this correlation, and equa-tions for calculating the positron annihilation charac-teristics. Section III contains the results and discussionfor the ideal Ga vacancy in GaAs as well as for the Gavacancy with self-consistent ionic relaxations. Section IVis a short summary.

II. CQMPUTATIONAI. METH&OS

A. Two-component density-functional theory

Our aim is to describe from first principles the systemof a positron trapped by a defect in a solid. This meansthat we need a calculation scheme, in which the electron

52 ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . . 10 949

and positron densities as well as the ion positions aredetermined self-consistently. This can be done using thetwo-component density-functional theory.

We write the total energy of the system as a functionalof the electron density n'(r), positron density n~(r), andthe ion positions I R~ j:

E [n', nJ', [RI }]=F[n', [RI j ]+F[n~, IRI j ]

+EH[n', nj']+E; ~[n', n~]

+E;,„(IRIj) .

Above, EH~[n', .n~] is the Coulomb (Hartree) interactionbetween the electron and positron densities andE; I' [n', n "] is the correlation energy between electronsand positrons. F [n, IRI }] are the one-component func-tionals for either the electron or the positron density, i.e.,

F [n I RI } ]=Eq;„[n]+E,„,[n, IRI } ]

+EH [n]+E„,[n],where E„;„[n],EH[n], and E„,[n] are the usual kineticenergy, Hartree energy, and exchange-correlation energyfunctionals of the one-component density-functionaltheory, respectively. E,„,[n, t RI j ] is the interaction en-

ergy between the charge density n and the nuclei Z~ atthe positions IRI j. The ground state of the system isfound by minimizing the total-energy functionalE [n ', n ~, I RI j ]. We do this by using for the total-energyfunctional a generalized Kohn-Sham form, in which theelectron density is obtained from the auxiliary one-particle wave functions P;(r):

Above, the summation is over all occupied electronstates. Because in the relevant experiments there is onlyone positron in the sample at a given time, the positrondensity corresponds to a single wave function

n ~(r ) =~

f"(r )

In practice, only the valence part n' of the electrondensity is calculated self-consistently. For the core elec-tron density n the free-atom distribution is used; i.e., it isassumed not to be affected by the solid environment or bythe presence of the positron. The interactions betweenthe valence electrons and the ion cores are describedwithin the pseudopotential model. The electron wavefunctions are presented by plane-wave expansions, thecoefficients of which have to be determined. This meansthat the part F[n', IR~ }]+E;,„(tRIj) of the total-energy functional is treated within the usual pseudopo-tential formalism explained, for example, in Refs. 14 and17. For the positron-ion core interactions core densitiesof free atoms are used. The positron wave function issolved in real space using a method' based on the conju-gate gradient optimization. ' The real-space point grid isthe same as used in performing the fast Fourier transfor-mations in the plane-wave approach for the valence elec-trons and this grid couples the valence electron and posi-

tron densities.Our calculation for a positron trapped by a defect

proceeds as follows. First the positions of the ions arekept fixed and the corresponding electron density is cal-culated without the effect of the positron. The FouriercoefFicients of the valence electron wave functions aredetermined by a steepest-descent minimization of thepure electronic part F[n', IRI j ] of the total-energy func-tional. Then the electron density is kept fixed and thepositron wave function is calculated in the points of thereal-space grid by minimizing the terms containingthe positron density, i.e., F[n~, I RI }]+EH [n ', n~]+E,' ~[n', n~]. Thereafter the electron density is iteratedby including also the cross terms containing the positrondensity. The whole electron-positron system is theniterated to self-consistency by repeating a sequence inwhich the positron density is updated always after everyelectronic iteration. When the self-consistency of theelectron and positron densities is achieved, the forces act-ing on the different ions I are calculated using theHellman-Feynman theorem as

F [n~, IRI j ] =E„;„[n~]+E,„,[n~, IRI } ] .

In the case of a positron trapped at a defect the SICmeans that the positron potential far from the defect is along-range Coulomb potential due to the extra electroncharge relaxed towards the center of the defect to screenthe positron. This long-range potential results in a morelocalized positron wave function than the short-rangeLDA potential.

The terms theof total-energy

The ions are moved according to these forces using adamped second-order dynamics algorithm. The electrondensity and the positron density following it are relaxedsimultaneously with the relaxation of the ions using ficti-tious dynamics for the coe%cients of the electron wavefunctions.

In the LDA of the density-functional theory for elec-tronic structure a localized electron interacts with itself,because the Hartree and exchange-correlation terms inthe energy functional Eq. (2) are calculated using the to-tal density. The situation is in a way most pathologicalfor a one-electron system, such as the hydrogen atom.The self-interaction in this case means that the attractive—1/r Coulomb potential due to the nucleus is substitut-ed at large distances by the short-range exchange-correlation potential. In the self-interaction correctionmethod (Ref. 19) the Hartree and exchange-correlationenergies of a given localized electron with itself are as-sumed to cancel each other. Then, for example, the po-tential in the hydrogen atom is corrected to the attractivelong-range —1 lr Coulomb potential. Because there isonly one localized positron in the systems considered inthis work, we explicitly make, following Boronski andNieminen, the self-interaction correction for the posi-tron. The functional I' for the positron density simplifiesthen as

10 950 M. J. PUSKA, ARI P. SEITSONEN, ANI3 R. M. NIEMINEN 52

E [n', n~, [Rl ] ] that do not contain the positron densityare treated within the usual pseudopotential formalism.The positron kinetic energy is calculated in the real-spacegrid using a five-point difference formula. The evaluationof the electrostatic terms containing the positron density

is briefly commented on below and the crucial treatmentof the electron-positron correlations is explained morecarefully in Sec. II B.

The electrostatic terms containing the positron densityin Eq. (1) are

where the electron density has been divided into valencen' and core n' parts. The integral involving the coreelectron charge and the corresponding part of thenucleus-positron interaction are performed in real space.The valence-electron-positron interaction is performed asa sum in the reciprocal lattice by first Fourier transform-ing the positron density calculated in real space. Actual-ly, the Fourier-transformed Hartree potential due to thevalence electron density and positive Gaussian chargedistributions describing the valence part of the nucleus isfirst calculated. Then, in the summation over the re-ciprocal lattice vectors the G=O term is excluded. Thisexclusion is done also in calculating the valence electron-ion interaction. Thus, in the case of a supercell with anominal net charge there will be an implicit neutralizingbackground charge and there are no long-range Coulombinteractions between the cells of the superlattice. Finally,the use of the Gaussian distribution instead of a pointlikeone is corrected by a summation in real space, in afashion similar to the ordinary Ewald summation.

The effective potentials needed in the electron and pos-itron Hamiltonians are obtained from the total energy asfunctional derivatives with respect to the electron or pos-itron densities. The effective potential for the electronsreads

+ E; ~[n '(r), n~(r) ],5

where V',z is the usual one-component effective electronpotential {containing in our scheme, e.g., the local andnonlocal pseudopotential terms). Similarly, the effectivepositron potential is

B. Electron-positron correlation energy

When a positron enters the electron system of a solid,it gathers a screening cloud of electrons around it. In thetwo-component formalism for a positron trapped by a de-fect the screening cloud consists of the relaxation of the"average" electron density n' towards the localized posi-tron density n& and of the "short-range" correlationcloud when the positron is thought to be fixed in a givenpoint in the system. In the total-energy functional thefirst effect of screening enters through the electron-positron Hartree interaction term in Eq. (1). The correla-tion part is included in the (unknown) functionalE; J'[n', n~]. Besides the correlation energy functionalthe correlation effects manifest themselves through thepositron-electron contact density (the so-called enhance-ment factor) used in calculating the annihilation rate.We will first consider the approximation for the correla-tion energy.

Boronski and Nieminen introduced the LDA for theelectron-positron correlation energy in the two-component theory by a function I',' ~ defining the correla-tion energy per unit volume at a given point. (In theone-component LDA one defines the exchange-correlation energy per particle. ) The arguments of F; ~

are the local electron and positron densities and the totalelectron-positron correlation energy of the system is ob-tained by integrating over the volume

E; ~(n', n ]=fF,' {n'(r),n (r))dr . (10)

The electron and positron correlation potentials, whichare defined as the functional derivatives of the correlationenergy with respect to the electron or positron density[see Eqs. (8) and (9)], are then simple partial derivatives ofthe function F; ~(n', n~):

BF; ~(n '(r), n ~(r) )V,'~(r) =

Bn

+ E; ~[n '(r), n~(r) ] .55n I'

These equations define the electron-positron correlationpotentials for electron ({5/5n')E; ~[n'(r), n~(r)]) andpositron ((5/5n~)E; ~[n'(r), n~(r)]) states

One reason for the few applications of the two-component density-functional formalism has been thelack of accurate interpolation forms for the electron-positron correlation energy in the LDA. The interpola-tion form presented by Boronski and Nieminen was

ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . .

based on symmetry arguments and on Lantto's' hyper-netted chain calculations for systems of equal electronand positron densities, and on a different type of many-body calculations by Arponen and Pajanne for the limitof a delocalized positron in a homogeneous electron gas.The scarcity of the data leads to difficulties in the con-struction of the interpolation form for correlation energyfor the whole electron-positron density plane. Theclearest manifestation of the difficulties is the nonmono-tonic behavior of the correlation potentials derived fromthe interpolation form of Boronski and Nieminen.

Lantto' has later published more data for theelectron-positron correlation energy, including several ra-tios of the electron and positron densities. On the basisof the these data we have constructed a new well-behaving interpolation form F,' (n', n ) Th.e ratio of theelectron and positron densities is denoted as

0.0

gj -0.5 .KE

—1.0

-1.5 .II VU.

-2.00 4 6 8

nI' (10 a.u.)10

FIG. 1. Electron-positron correlation energy per unit volume

F, ~(n', n~). The energy is shown as a function of the positrondensity for various values of the electron density. The figure is

based on the interpolation form of Eq. (14).

x =ni'ln' . (12)

Lantto has given the correlation energy for eight x valuesin the range x =0, . . . , 1 for electron densities withr, =1, 2, 3, 4, and 5 a.u. We require that the functionF,'~(n', n~) is symmetric with respect to the change ofthe positron and electron densities. Furthermore, we re-quire that in the limit of vanishing positron density theform

E; J'[n ', n i'] =E; i' [n ', n ~~0]= In~(r)eAp(n'(r))dr,

is obtained. Above, ezp(n) is the correlation energy for adelocalized positron in a homogeneous electron gas. Dueto symmetry, a corresponding form is required on thelimit of vanishing electron density. We have used fore~p( n ) the results calculated by Arponen and Pajanneas parametrized by Boronski and Nieminen.

For the interpolation we use the functional form

1 =a(r,')+b(r,')rf +c(r,')(r~)E'I'(n' n~)

The above fit behaves well around the typical electronand positron densities at vacancies in semiconductors(r,'=r~=3 a.u.). The fit is, however, based on resultswith r, ~ 1 a.u. , only. In the ion-core region where r,' isless than unity and r~ is very large, one should in practi-cal calculations of the correlation energy smoothly switchto the form. of Eq. (13) for the vanishing positron density.We use the matching point of r,'=0.7 a.u. The small er-rors introduced to the positron correlation energy in thissubstitution are not important, because in the high-density ion core region the positron wave function is verysmall. The errors in the positron correlation potentialare not important, because the Coulomb repulsion dom-inates over the correlation.

The resulting electron-positron correlation energy den-sities are shown in Fig. 1 as a function of the positrondensity. Although the qualitative behavior of the curvesin Fig. 1 is the same as that in the corresponding figuregiven by Boronski and Nieminen there are important

A, =69.7029, B,= —107.4927, C, =23.7182,(16)

B,= 141.8458, Cb = 33.6472, C, =5.21152 .

+ --+~Ap( r ) ~Ap( r

where r,'(r~) is the electron (positron) density parameterdefined by n'= /3[4m(r, ') ] (n~=3/[4~(r~) ]). The sym-metry with respect to r~ and r,' means that the functionsa (r,'), b (r,'), and c(r,') are second-order polynomials,

a(r,')= 3,+B,r,'+C, (r,')

b (r,'=B, +Bbr,'+Cb(r, ')

c(r,')=C, +Cbr,'+C, (r,')

The coefficients are determined by the linear least-squaresfit to Lantto's data. ' When the densities and the r, pa-rameters are given in a.u. and the correlation energy in

Ry the coefficients have the values

0.0CD

-1.0-I-KLLI e-2.0- I'

QI0 -30-I-

K~ -4.0-K0~ -50

0

yee

4I'~ (a.u. )

FIG. 2. Electron-positron correlation potentials for electronsV; ~(n', n~) and positrons V~ ~(n', n~). The potentials are shown

as a function of the positron density parameter for variousvalues of the electron density. The figure is based on the inter-

polation form of Eq. (14).

10 952 M. J. PUSKA, ARI P. SEITSONEN, AND R. M. NIEMINEN

quantitative differences between the two results. Themain reason for the differences can be traced back to thescarcity of the original data used in Ref. 9. The electronand positron parts of the electron-positron correlationpotential calculated according to Eq. (11) from the corre-lation energy are shown in Fig. 2. For a fixed electrondensity, the positron part rises rapidly as the positrondensity increases (to the left in the figure). This meansthat the number of electrons becomes more and moreinsufhcient to screen the positron density. The presentpotentials are more repulsive than those given in Ref. 9and, for example, the positron potentials decay towardsthe results of the vanishing positron limit more slowlythan in the old interpolation results.

C. DifFerent schemes for calculatinglocalized positron states

The use of the electron-positron correlation energy inthe LDA [Eq. (10)] and the corresponding determinationof the annihilation characteristics (Sec. II D below) con-stitutes the two-component BN calculation scheme. Asmentioned in the Introduction, for the description of thelocalized positron states there exists also the CONVscheme relying on the normal one-component density-functional theory. Moreover, the GGGC schemerepresents a different type of two-component approach.In this subsection we describe the calculation of the elec-tron and positron states in these schemes. The descrip-tion of the calculation of the annihilation characteristicsis postponed to Sec. II D.

In the case of a delocalized positron in a perfect crystallattice the positron density is vanishingly small at everypoint. Therefore the electron density is not affected bythe positron and can be calculated as in the usual one-component formalism. From the electron density, thepositron potential is constructed in the limit of the van-ishing positron density in accordance with Eqs. (9) and(13) as

V"(r)= Vc „(ri)+e~ (pn (r}), (17)

where the Coulomb part Vc,„& arises from the nuclei andthe electron density. Then the positron wave function issolved. No self-consistency iterations are needed whenintroducing the positron.

The procedure for delocalized positron states is re-markably simpler than the full two-component schemerequired for the localized positron states. For this reasonand also because of the lack of the two-componentelectron-positron correlation functional the procedurehas been widely used also for localized positron states.This procedure is the above-mentioned CONV scheme.It can be justified by arguing that the positron togetherwith its screening cloud enters the system as a neutralquasiparticle, which does not affect the average electrondensity. The two-component and the CONV schemescan be (to some extent) viewed as two different ways todivide the total positron potential in to the correlationpart and the Coulomb part: In the CONV scheme theeffects of the positron screening are taken into accountonly via the electron-positron correlation potential

0.0Q K

-2.0-I-

I- -4.0-'g.'0CL

0 -60-I-

K~ -80-K0

-10.00

I I

4 6r~ (a.u.)

FIG. 3. Electron-positron correlation potentials for electronsV, ( n ', n ) and positrons Vp ( n ', n ). The potentials are shownas a function of the positron density parameter for the electrondensity parameter «,'=3 a.u. The solid lines are based on the in-

terpolation form of Eq. (14) whereas the dashed lines corre-spond to the limit of vanishing positron density, Eq. (13).

whereas in the two-component theory they contribute,due to the relaxation of the electron density, also via theHartree term. In the two-component theory the correla-tion part decreases in magnitude when the positron densi-ty increases and this opposes the lowering of the Hartreepart due to the increase of the electron density relative tothe CONV scheme (near the regions where the positrondensity is high). As a result, as will be seen below, thepositron density and the ensuing positron annihilationcharacteristics calculated with the CONV and the BN-LDA scheme are very similar (if the ion positions arekept fixed). The differences in the annihilation charac-teristics are damped also because the increase of the an-nihilation rate due to the increase of the average valenceelectron density near the localized positron is compensat-ed due to the decrease of the (correlation) contact densityas the positron density increases.

In the two-component GGGC scheme the positron po-tential does not depend on the positron density. This isachieved by using in the LDA a two-componentelectron-positron correlation energy in the limit of van-ishing positron density [Eq. (13)] irrespective of the actu-al value of the positron density. The model makes theimplementation of the two-component scheme easy, be-cau.se for a given electron density the positron state canbe directly calculated without (positron) self-consistencyiterations that are necessary in the BN scheme. Themodel can be justified' also by the fact that it is freefrom positron self-interaction in the sense that the posi-tron potential does not depend on the positron density atall (in the BN scheme the positron potential depends onthe positron potential via the electron-positron correla-tion potential). However, the GGGC scheme clearlyoverestimates the electron-positron correlation effects:The positron potential is lowered via the Coulomb (Har-tree) term and at the same time the positron correlationpotential is calculated so that it has the maximumstrength, which appears for the vanishing positron densi-ty. According to Fig. 3, for the electron and positrondensity parameters r~ =r,'= 3 a.u. corresponding to

52 ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . . 10 953

roughly the situation in a semiconductor vacancy, thepositron correlation potential is almost 6 eV lower in theGGGC scheme than in the BN scheme with the new in-terpolation function. This leads to very localized posi-tron states at defects. Gilgien et al. ' report even self-trapping of the positron in the interstitial region of theperfect GaAs lattice. Mobility measurements '

by theslow-positron-beam technique clearly rule out positronself-trapping.

D. Positron annihilation characteristics

In the two-component LDA calculations the positronannihilation rate as a function of the momentum p of theannihilating positron electron pair is determined as

p(p)=mr, c g f e' F'g(r)P'(r)

X '1/g(n '(r), nF(r) )dr (18)

—1.4820r, +0.3956r, ~ +r, /6, (19)

where r, is the classical electron radius and c is the speedof light. The summation is over all occupied electronstates. For the enhancement factor (the contact density)g ( n ', n F ) we use the functional form introduced byBoronski and Nieminen for interpolating in the plane ofelectron and positron densities. The form includes thefunction go for the limit n~ —+O,g, for the case of n~=n',and gz for the case of nF=n'/2. We have obtained thecoefficients of these functions by fitting Lantto's dataquoted by Boronski and Nieminen. The resulting func-tions are (r, in a.u. )

go ( r, ) = 1+1.23r, +0.98890r,

in solids the annihilation rate is calculated on the limit ofthe vanishing positron density, i.e., also the contact elec-tron density is treated as for delocalized positrons and theenhancement function go(r, ) is used. The CONV schemehas been in wide use and it gives positron lifetimes at va-cancies that are in a reasonable agreement with the ex-perimental results for metals and also for semiconduc-tors, if reasonable ionic relaxations around the vacancyare assumed. '

In the two-component GGGC scheme the enhance-ment function go(r, ) on the limit of the vanishing posi-tron density is used. Because this enhancement factor islarger than the two-component value with the same elec-tron density but with a finite positron density, the ap-proximation shortens the positron lifetime. This acts tocancel the tendency of the model to increase the positronlifetime via the strong positron localization.

The magnitude of the core annihilation rate is moni-tored by the so-called core annihilation or 8' parameterof the line-shape measurements. It is defined as the ratioof the annihilations occurring inside a given high-momentum window to the total number of annihilations.Because at high enough momenta only the annihilationswith core electrons contribute, the changes in the 8'pa-rameter reAect changes in the relative intensity of thecore annihilation and thereby changes in the positron-core-electron overlap. Usually one calculates the ratiobetween the 8'parameter for a sample containing a cer-tain defect type and the 8'parameter corresponding tothe perfect crystal lattice. This ratio is called the relative8' parameter. The relative 8' parameter can be estimat-ed from the theoretical core annihilation momentum dis-tributions. However, an approximation based on the to-tal core annihilation rates already gives a good accountfor its magnitude. When integrated over momentum, theshape parameter 8'can be written as

g &(r, ) = 1+2.0286r, —3.3892r, ~

+3.0547r, —1.0540r, +r, /6,

g2(r, ) = 1+0.2499r, +0.2949r, ~

(20)

/g )DefectC

( g /g )Free(24)

+0.6944r, —0.5339r, +r, /6 . (21)

These interpolation forms differ slightly from the fitsmade by Boronski and Nieminen.

The total annihilation rate is obtained from Eq. (18) byintegrating over the momentum. The result is

i=mr, c f n~(r) 'n(r)g(n'(r), n~(r)) rd.

A,, =f n~(r)g(n'(r), n~(r))n'(r)dr,(23)

A,, = f n (r)gF(n'(r), (rn)F)n'(r)dr .

In the CONV scheme for localized positrons at defects

In the LDA for positron annihilation the total annihila-tion rate is divided into core (A,, ) and valence (A, , ) contri-butions as

where k, and k are the annihilation rate with core elec-trons and the total annihilation rate, respectively, calcu-lated for a positron trapped at the defect or for a free pos-itron. In this work we use the LDA model of Eqs. (22)and (23) for A, and A,„respectively.

The positron trapping rate, at which the transitionform the delocalized positron state to the localized onetakes place, is an important experimental parameter. Itstemperature dependence gives information about thecharge state of the defect. Its magnitude gives an esti-mate for the defect concentration, provided that the trap-ping coefficient, i.e., the trapping rate per unit defect con-centration is known. Theoretically, the type of trappingprocess, with a certain mechanism of the energy transferto the host, depends crucially on the amount of energyreleased by the positron in the transition from the delo-calized to the localized state. This is true also for themagnitude of the trapping rate. The energy release in theprocess is called the trapping energy. In the CONVscheme the trapping energy is calculated as the difference

M. J. PUSKA, ARI P. SEITSQNEN, AND R. M. NIEMINEN

of the positron energy eigenvalues for the bulk and va-cancy systems. In the two-component formalism one hasto consider the whole total-energy functional. The trap-ping energy is then

E, =E„,(bulk+e+) —E„,(bulk)

—[E„,(defect+ e+ )—E„,(defect) ],

where the terms on the right-hand side refer to the totalenergy of the perfect bulk lattice with a positron, thatwithout a positron, the total energy of the defect with apositron, and that of a defect lattice, respectively. Theenergy diff'erence E„,(bulk+e+) —E„,(bulk) is equal tothe positron energy eigenvalue in the bulk for the CONVscheme as well as for the GGGC scheme.

III. RESULTS AND DISCUSSIGN

We demonstrate the calculation scheme in the case ofthe III-V compound semiconductor GaAs, consideringthe Ga vacancy in the triply negative charge state. Theelectron-ion interactions are described by norm-conserving pseudopotentials. The omittance of Ga, e-states and the overlap of core and valence electron densi-ties are (partially) corrected using the nonlinear core-valence exchange-correlation scheme. Besides the localpart (d component) the pseudopotentials contain nonlo-cal s and p components. In the plane-wave expansionsfor the electron states we use the cutoff energy of 16.8Ry, which corresponds to about 11000 plane waves pereigenstate. We use a cubic supercell with periodic bound-ary conditions. The supercell contains 64 (63) atoms inthe case of the perfect bulk lattice (the Ga vacancy).Only the I point is used to sample the first Brillouinzone. In calculating the relaxations of the ions aroundthe vacancy with molecular dynamics we use the truephysical masses of the ions and the fictitious mass of 400a.u. for the electronic states. The time step of 6 a.u. isused in solving the equations of motion.

Our calculations for the perfect bulk lattice gives thelattice constant of S.62 A, which is in good agreementwith the experimental value of 5.65 A. The positronbulk lifetime calculated in the CONV scheme is 214 ps.By construction, this is the bulk lifetime for the BN andGGGC two-component schemes also. The positron bulklifetime calculated without the electron-positron correla-tion potential (in the NOCORR scheme) is, due to the re-duced annihilation near the ion ones, a somewhat longer227 ps. The calculated positron lifetimes are shorter thanthe experimental value of 231 ps (Ref. 15). This is partlydue to the smaller lattice constant, but the main reason isthat the LDA in calculating the positron annihilationrate leads to generally too short positron lifetimes. Theerror is, for a wide range of materials and positron life-times, proportional to the positron lifetime itself. There-fore a simple scaling [in the case of the interpolation formby Boronski and Nieminen the scaling factor is —1.1

(Ref. 28)] can be used to correct the I.DA lifetimes toagree well with experiments. In this work, we will there-fore scale positron lifetimes for vacancies by the factor of231/214=1.08 with the exception of the values obtained

without the correlation potential for which the scalingfactor is 231/227 = 1.02.

A. Ideal Ga vacancy in GaAs

8-4

12 -b)BL1r

BL

QeL

12 —g)

BL

Qzl%r

[11 I] DIRECTION

FIG. 4. Electron and positron densities in the ideal triplynegative Ga vacancy in GaAs. The results are shown for theCQNV scheme (a), and for the two-component BN (b), GGGC(c), and NOCQRR schemes (d). The electron (positron) densityis given by a solid (dashed} curve along the line in the [111]direction through the center of the vacancy (open circle) and Gaand As atoms (filled circles).

First we want to compare the results of the differentschemes for calculating the positron annihilation charac-teristic for a defect trapping a positron. In order to sim-plify the comparison we have solved for the self-consistent electron structure and the positron state for anideal triply negative Ga vacancy (Vo, ) in GaAs. Theionization levels of the Ga vacancy lie in the lower half ofthe band gap, and therefore the triply negative chargestate corresponds to the experimental situation in semi-insulating and n-type GaAs. The ideal vacancy meansthat the ions are not allowed to relax from their ideal lat-tice positions.

The electron and positron densities obtained in thedifferent schemes are compared in Fig. 4, which showsthem on the line going through the center of the vacancyalong the [111]direction. A similar comparison for thetotal positron potential and its Hartree and correlationcomponents is given in Fig. 5. Figures 4(a) and 5(a) cor-respond to the CONV scheme, in which the electron den-sity is calculated without the inhuence of the positronand the positron state is determined without self-consistency iterations. In Figs. 4(b) and 5(b) the resultsobtained in the two-component BN scheme and in Figs.4(c) and S(c) those of the GGGC scheme are shown. Fi-nally, Figs. 4(d) and 5(d) show the densities and potentialsfor the NOCORR scheme. Comparing the two upper-

52 ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . . 10 955

o.o -)-0.4 —::-'

-0.Sic

-0.4 -::"

C5

~ -0.8ir'

p OO-)I-O -0.4 -::-'..

b)

-0.8ir'%r

0.0—

-0.4—

AW

[1111DIRECTION

FIG. 5. Positron potential in the ideal triply negative Ga va-cancy in GaAs. The results are shown for the CONY scheme(a), and for the two-component BN (b), GGGC (c), and NO-CORR schemes (d). The total potential (solid curve) with itsdecomposition into the Hartree (dashed curve) and the correla-tion (dotted curve) are shown along the line in the [11 lj direc-tion through the center of the vacancy (open circle) and Ga andAs atoms (filled circles).

most panels of Fig. 4 it can be seen that the average elec-tron density at the vacancy clearly increases from Fig.4(a) to Fig. 4(b), but the positron densities are closer toeach other. In the GGGC scheme the increase of theelectron density is according to Fig. 4(c) somewhatstronger than in the BN scheme, but the maximum posi-tron density is by a factor of two larger than in theCQNV or in the BN scheme. The positron density ob-tained in the NOCORR scheme is according to Fig. 4(d)nearly the same as in the GGGC scheme. The electrondensity at the vacancy is slightly lower in Fig. 4(d) thanin Fig. 4(c) due to the omission of the attractive electronpart of the electron-positron correlation.

Figure 5, which shows the positron potentials, explainsnicely the trends seen in the positron density. In thetwo-component BN scheme the total positron potential isquite similar to that in the CONV scheme. This is a re-sult of the cancellation of the changes in the Hartree po-tential and in the correlation potential when the modelfor the positron screening changes from the conventionalone to that of the two-component theory. It can be seenthat the Hartree potential is lowered when the electrondensity at the vacancy increases, but the two-componentcorrelation potential rises when the positron density in-creases. In the GGGC scheme this cancellation does notoccur, because, to a rather good approximation, thecorrelation potential is the same as in the CQNV schemewhereas the Hartree potential is close to that obtainedalso in the BN scheme.

In the two-component NOCORR calculation the total

potential of the positron consists only of the Hartreeterm, which to a high accuracy is the same as that ob-tained in the GGGC scheme. Because the correlation po-tential in the GGGC scheme is quite structureless insidethe vacancy, its omission does not affect the positron den-sity. In the CONY scheme the omission of the positroncorrelation potential improves the positron localizationat the vacancy, because due to the higher electron densitythe correlation potential is more attractive in the intersti-tial regions than at the vacancy. When the electron den-sity is then allowed to relax in a two-component calcula-tion the localization is further improved due to the accu-mulation of the screening electron charge at the vacancy.

Figure 6 shows the contour map of the relaxation ofthe electron density due to the introduction of the posi-tron into the ideal Ga vacancy. This map is obtained as adifference between the electron density calculated withthe positron using the BN scheme and that of a clean va-cancy. The electron density decreases quite evenly fromthe bond regions far from the vacancy. Close to the va-cancy the screening charge shows maxima at the dan-gling bonds. Thus, the positron shows a tendency to-wards to healing the effects of the vacancy to the bulkelectronic structure.

The positron lifetimes, core and valence annihilationrates, as well as the trapping energies at the ideal triplynegative Ga vacancy calculated in different schemes areshown in Table I. The lifetimes and partial annihilationrates are scaled so that the bulk lifetime coincides withthe experimental one of 231 ps (see the discussion above).The CONV scheme calculation as well as the two-component BN and GGGC schemes give quite similarlifetimes, 270—278 ps. Also the calculation with the NO-CORR scheme gives a similar result, 264 ps, if theenhancement is taken in the limit of the vanishing posi-tron density. The fact that the CONV scheme and theGGGC scheme give practically the same positron life-time is a result of a feedback effect: Both schemes usethe same functions for the construction of the positronpotential and the annihilation rate. Then, if the electron

0I-

%F

[110]DIRECTION

FIG. 6. Electron density induced by a positron trapped bythe ideal triply negative Ga vacancy in GaAs. The two-component scheme by Boronski and Nieminen is used. Thefigure shows a region of the (110)plane limited by the borders ofthe simulation cell. The contours of the positive (negative)values are shown as solid (dashed) curves. The contour of thezero density is denoted by a bold solid line. The contour spac-ing is one-sixth of the maximum value.

10 956 M. J. PUSKA, ARI P. SEITSONEN, ANIL R. M. NIEMINEN 52

TABLE I. Positron lifetime ~, core annihilation rate A,„valence annihilation rate k„relative 8'pa-rameter [Eq. (24)], and trapping energy E, at the ideal triply negative Cia vacancy in CiaAs. The resultsof the CONV scheme, the two-component calculations within BN, GGGC, and NOCORR schemes aregiven. The lifetimes and annihilation rates are scaled so that the theoretical bulk lifetime coincideswith the experimental one of 231 ps (Ref. 15). In the two last cases the lifetimes given in theparentheses are obtained using the two-component form of Eq. (22) depending on both the electron andpositron densities. The annihilation rates for the perfect bulk lattice calculated using the correlationpotential (CONV) or without using it (NOCORR) are given for comparison.

Scheme

Ideal vacancyCONVBNGGGCNOCORRBulkCONVNOCORR

~ (ps)

270278

270 (311)264 (308)

231231

A,, (1/ns)

0.2790.2920.1540.135

0.5540.468

A, „(1/ns)

3.4223.3043.5463.649

3.7753.861

0.590.650.320.33

E, (eV)

0.730.602.462.64

density changes, due to approximations used in its con-struction or in the present two-component formalism dueto the Gnite positron density, the positron density followsthe electron density in such a way that the electron-positron overlap and the annihilation rate are conserved.In the present case the strength of the feedback e6ect isobvious because the changes in the electron and especial-ly in the positron densities between the CONV and theGGGC methods are very large (see Fig. 4). The similari-ty of the lifetimes of the CONV and the BN schemes is aresult of the near cancellation of the increase of the an-nihilation rate due to the increased electron density andthe decrease the annihilation rate due to the finite posi-tron density when switching from the former scheme tothe latter.

According to Table I, the annihilation rate for the coreelectrons is much more sensitive to the calculationscheme than the positron lifetime. The CQNV and theBN schemes give for the ideal Ga-vacancy practically thesame core annihilation rate of -0.3 ns '. The GGGCand NOCORR schemes gives core annihilation ratessmaller by approximately a factor of two. The reductionseen in the GGGC and NOCORR schemes relative to theCONV and the BN schemes is due to the stronger posi-tron localization in the former. The relative 8'parame-ters, calculated from Eq. (24) using the annihilation ratesof the CQNV and BN schemes give similar results, i.e.,0.59 and 0.65, respectively. The values from the GGGCand NOCORR are remarkably lower, 0.32 and 0.33, re-spectively. The relative 8' parameter measured for theGa vacancy by the standard Doppler broadening tech-nique is 0.89 (Ref. 30), while high-resolution coinci-dence-Doppler measurements give values between 0.7and 0.8 (Ref. 31). Thus, these experiments strongly sup-port the CONV and the BN schemes in the comparisonwith the GGGC and NOCORR schemes. This con-clusion is even strengthened when the relaxation of theions around the vacancy is considered below.

We have also calculated the positron lifetimes by usingthe electron and positron densities of the GGGC scheme

but employing the two-component enhancement factor.This kind of calculation is also done with densities ob-tained with the NOCORR scheme. The resulting life-times, which are shown in parentheses in Table I, are re-markably long. This is a result of the strong positron lo-calization at the open volume of the vacancy. The in-crease of the positron lifetime is opposed in the GGGCand NOCQRR schemes when the enhancement at thelimit of the vanishing positron density is used with the in-creased (relaxed) electron density.

The experimental positron lifetime for the (triply nega-tive) Ga vacancy is 260 ps. ' In order to reproduce thisthe ions neighboring the vacancy should relax inwards inall the schemes considered. The above comparison of thetheoretical relative 8' parameters with the experimentalones also support this conclusion. The CONV and thetwo-component BN schemes give similar, relatively small(0.6—0.7 eV) trapping energies. They are in agreementwith previous estimates, ' '" which are typically smallerthan those for metal vacancies. The GGGC and theNOCORR schemes give substantially (by -2 eV) largerpositron trapping energies.

Next we consider the 2D-ACAR spectra calculatedfrom the valence electron densities in the diA'erentschemes. The 2D-ACAR maps corresponding to the per-fect bulk lattice and the ideal triply negative Ga vacancycalculated in the CONV scheme are shown in Fig. 7. Themomentum density p(p) of Eq. (18) is integrated alongthe [110] direction. The characteristic feature of thebulk distribution is the anisotropy with "shoulders"pointing to the [111]directions. The distribution for thevacancy is narrower due to the increase of the relativecontribution of the annihilations with the low-momentumvalence electrons. Also the shoulders have disappearedand only a slight anisotropy remains so that the contoursreach further from the origin in the [110] direction thatin the [100] direction. A comprehensive analysis of themomentum distributions for the bulk semiconductors aswell as for their vacancies is due to Saito, Oshiyama, andTanlgawa.

52 ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . . 10 957

The cuts of 20-ACAR spectra for the ideal Ga vacan-cy are shown in Figs. 7(c) and 7(d) along the [100] and[110]directions, respectively. All the curves are normal-ized to unity at zero momentum. The results calculatedin the difterent schemes are compared with the experi-mental ones. ' The narrowest and also the most isotropicdistribution results in the GGGC scheme rejecting againthe strong localization of the positron wave function atthe vacancy. The CONV scheme gives a distribution thatis, especially along the [110]direction, close to that of theGGGC scheme. The BN scheme gives results in the bestagreement with the experiment. However, all thetheoretical results for the ideal vacancy are too narrowand isotropic in comparison with the experimental 20-ACAR distribution. The inward relaxation of the ionsaround the vacancy would again improve the situation bychanging them towards the bulk result, which has a re-markable anisotropy [Fig. 7(a)].

B. Ion relaxations around a triply negativeGa vacancy in GaAs

In order to understand the efFects of ion relaxation onthe positron annihilation characteristics we have first per-formed calculations in which only the first neighbors of

7.5

tge

E

ID

& 0.0)KQ

e

LLI

5X

the vacancy are moved from their ideal lattice positions.In these calculations the ions are frozen to positions cor-responding to a given amplitude of the breathing-moderelaxation conserving the Td symmetry group of the idealvacancy. This is consistent with the symmetry of the ful-ly relaxed Vz, vacancy (see below). The total energies[Eq. (1)] of the electron-positron system calculated in theBN and GGGC schemes are given in Fig. 8(a) as a func-tion of the amplitude of the relaxation. The CONVscheme results included in Fig. 8(a) refer to the model"in which the total energy, is the sum of the pure electronicenergy without the inAuence of the positron and the posi-tron energy eigenvalue calculated within the CON Vscheme. The results of the BN scheme and the CONVscheme nearly coincide. They show an energy minimumwhen the nearest-neighbor ions are moved inwards a dis-tance, which corresponds to about 2% of the ideal latticebond length. The similarity of these curves reAects thesimilarity of the positron energetics in these schemes. Inthe GGGC scheme the energy minimum is found to hap-pen without any relaxation of the nearest-neighboratoms. Especially in the CONV and BN schemes thetotal-energy curve is quite Aat. The faatness would beeven more pronounced if the ions beyond the nearestneighbors were allowed to relax and lower the total ener-gy. The fact that the electronic total energy is a decreas-ing function of the inward relaxation whereas the posi-tron energy eigenvalue rises when the open volume at thevacancy decreases is an important reason for the slow

0.4 —;

Ig 0.2-KLU

LLJ

1

-S.s 0.0 7.5 -7.5 0.0 7.5ANGLE ALONG [110](mRad) ANGLE ALONG [110](IRad)

1.0

aXl

0.5KU

ICl

0.0290

270-CL

LLI

250-I-LL.

2301.0

IX.'

I- 0.8

e 0.6KQ. 0.4

0

b)

0 0 5 ~ ~ ~

0 5ANGLE (mRad)

10 0ANGLE (nRad)

10 0.2 -10I

-5RELAXATION (%)

FIG. 7. Angular correlations of annihilation photons in-tegrated along the [110] direction. The results for the perfectlattice (a), as well as for the ideal triply negative Ga vacancy inthe conventional scheme (b) are shown. The contour spacing isone-tenth of the maximum value. The cuts along the [100] (c)and [110] (d) directions are given in the case of the Cra vacancy.The experimental results (Ref. 16) are given in (c) and (d) bysolid lines, the results of the CONV (open circles), BN (solid cir-cles), and GGGC (crosses) schemes are given at the discretepoints for which the calculations are performed.

FIG. 8. Total energy (a), positron lifetime (b), and relative 8parameter (c) for the triply negative Ga vacancy in GaAs trap-ping a positron. The results are given as functions of thebreathing-mode relaxation of the ions neighboring the vacancy.The more distant ions are kept at the ideal lattice positions.The amplitude of the relaxation is given in percent of the bulkbond distance and negative (positive) values correspond to in-

ward (outward) relaxation from the ideal lattice positions. Theresults of the BN (solid line), GGGC (dashed line), and CONV(open circles) schemes are shown.

10 9S8 M. J. PUSKA, ARI P. SEITSONEN, AND R. M. NIEMINEN

variation of the total energy as a function of the ion re-laxation. The Aatness of the total energy means that thedetermination of the self-consistent ionic relaxations maybe dificult, because the result may depend strongly onthe approximations made.

The positron lifetime and the relative 8 parameter[Eq. (24)] are shown in Figs. 8(b) and 8(c), respectively.The results of the di6'erent schemes as a function of therelaxation of the nearest-neighbor ions of the vacancy aregiven. The positron lifetimes in the CONV and in theGGG-C schemes are nearly identical due to the feedbackmechanism explained above. The positron lifetime in-creases linearly when the ions relax outwards with theslope of about 3.S ps/%. This is in accord with the previ-ous theoretical estimations for the As vacancy in GaAs(Ref. 8) and with those for the vacancy in Si. The slopein the BN scheme calculations is slightly larger, about 4.5ps/%. The relative W parameter increases when the ionsneighboring the vacancy relax inwards. The 8' parame-ters obtained in the CONV and BN schemes agree withthe values, 0.7—0.8, of the coincidence Doppler experi-ment ' if the ions neighboring the vacancy have relaxedslightly inwards. The W parameter obtained in theGGGC scheme is in clear disagreement with the experi-ment for all reasonable values of the relaxation.

Next we have allowed all the ions in the supercell to re-lax to the configuration corresponding to the minimum ofthe total energy. First the ionic relaxations and the elec-tronic structure of the triply negative Ga vacancy are cal-culated without the trapped positron. Because all the lo-calized electron states in the band gap are occupied, thereis no strive for a symmetry-lowering Jahn-Teller relaxa-tion and the relaxed vacancy conserves the T& symmetrygroup. Thus the relaxation is of the pure breathing-modetype, and the atoms neighboring the vacancy relax in-wards to the center of the vacancy by 10.7% of the ideallattice bond length. This magnitude of the relaxation isclearly larger than the inward relaxation of about 4%found by Laasonen, Nieminen, and Puska using a simi-lar molecular-dynamics technique. However, the com-parison of these results is not straightforward, for exam-

pie, due to the di6'erent pseudopotentials used and thediferent lattice constants obtained. For comparison, thetight-binding calculation by Seong and Lewis has givenan inward relaxation of about 13%. The inward relaxa-tion found in the present calculation is so strong thatwithout the positron-induced relaxation the vacancy doesnot sustain a well-localized positron state neither accord-ing to the CONV scheme calculations nor according toany of the two-component schemes.

We then allow the ions around the vacancy to relaxdue to the introduction of the positron into the system.The resulting breathing-mode relaxations as well as thepositron annihilation characteristics for the difFerenttwo-component schemes are given in Table II. In allschemes the positron induces an outward ionic relaxationrelative to the clean relaxed vacancy. According to theBN scheme the positron-induced relaxation is -4% ofthe bond length and the nearest-neighbor ions of the va-cant site end up at positions that are a distance of 6.6%of the bond length inwards from the ideal lattice posi-tions. This inward relaxation is stronger than that corre-sponding to the total-energy minimum in Fig. 8(a) be-cause in the latter case ions beyond the nearest neighborsare not allowed to relax and thereby they hinder the re-laxation of the nearest-neighbor atoms. In the GGGCscheme the effect of the localized positron is larger, thenearest-neighbor ions of the vacancy site end up at posi-tions corresponding to an inward relaxation of 1.6% ofthe bond distance from the ideal lattice positions. Thepositron in this model is able to nearly heal the atomicpositions around the vacancy to those of the ideal vacan-cy.

The inward relaxation found in the BN scheme isslightly too large in magnitude to reproduce the experi-mental positron lifetime. The positron wave function isnot localized enough and the (scaled) positron lifetime ob-tained, 240 ps, is only slightly longer than the positronbulk lifetime of 231 ps. The 8' parameter has the valueof 0.88, which i.s larger than those determined by thecoincidence Doppler technique indicating also that thepositron wave function is not localized enough. The posi-

A, , (1/ns)

TABLE II. Self-consistent solution for the electronic structure, positron state, and the ionic relaxa-tions for the triply negative Ga vacancy in GaAs. The positron lifetime ~, core annihilation rate A.„valence annihilation rate A,„relative W parameter IEq. (24)], the trapping energy E„and the amplituded of breathing-mode relaxation relative to the bond length are given for the two-component BN,GGGC, and NOCORR schemes. BN4 refers to a calculation in which the nearest-neighbor ions of thevacancy are relaxed 4% of the bond length inwards from their ideal lattice positions and all the otherions are kept fixed. The lifetimes and annihilation rates are scaled so that the theoretical bulk lifetimecoincides with the experimental one of 231 ps (Ref. 15). In the case of the GGGC and NOCORRschemes the lifetimes given in the parentheses are obtained using the two-component form of Eq. (22)depending on both the electron and positron densities. The positive (negative) sign of d means an out-ward (inward) breathing-mode relaxation.

Scheme ~ (ps) A,, (1/ns) E, (eV) d (%)

BNGGGCNOCORRBN4

240268(306)272(308)

0.4680.1620.1240.354

3.6843.5733.5563.498

0.880.340.310.72

0.091.852.040.27

—6.6—1.6+3.0—4.0

ELECTRON-POSITRON CAR-PARRINELLO METHODS: SELF-. . . 10 959

tron trapping energy is consistently relatively small, only0.09 eV. On the contrary, the GGGC scheme results intoo localized a positron state. Although the calculatedpositron lifetime of 268 ps is only slightly longer than theexperimental one of 260 ps, ' the calculated 8'parameteris by a factor of two too small, i.e., 0.34, indicating amuch too small overlap of the positron with the coreelectrons. The positron trapping energy is also high, 1.9eV. The NOCORR calculation gives again very similarresults to the GGGC scheme. The positron-induced re-laxation is slightly larger in the NOCORR scheme result-ing in a somewhat longer positron lifetime, a smaller 8'parameter, and a larger positron binding energy. If thetwo-component enhancement, which depends also on thepositron density, is used in the GGGC and NOCORRschemes, the positron lifetimes are longer than 300 ps.

The 3D-ACAR maps calculated in the BN and GGGCschemes for the self-consistently relaxed triply negativeGa vacancy are compared with the experimental one inFig. 9. Compared with the experiment the GGGCscheme gives too narrow and isotropic a distribution,rejecting again the very localized nature of the positronwave function. The map corresponding to the BNscheme is more anisotropic, showing some bulklikefeatures [compare with Fig. 7(a)], which is a signal of asomewhat too delocalized positron wave function.

The comparison of the calculated positron lifetimes,the relative 8'parameters, and the 2D-ACAR maps withthe corresponding experimental ones indicates that theGGGC scheme results in too localized positron wavefunctions. In the first place, this is a result of the con-struction of the positron potential in the scheme; impos-ing a stronger inward relaxation of the ions neighboringthe vacancy does not remedy this character as can be bestseen in the behavior of the relative 8'parameter in Fig.8(c). On the other hand, the BN scheme with the fully re-laxed ion positions gives slightly too extended positronwave functions. This character, however, can beremedied by small changes in the ionic relaxation. Forexample, Figs. 8(b) and 8(c) indicate that the system, inwhich only the nearest-neighbor atoms of the Ga vacancyhave relaxed inwards about 4% of the bond length, givesin the BN scheme a positron lifetime and relative 8'pa-rameter, which are in good agreement with experimentalvalues (the calculated values obtained are also given inTable II). This same system gives a 2D-ACAR map thataccording to Figs. 9(d) and 9(e) compared well with themeasured one. The reason that the BN scheme does notgive ion positions that reproduce accurately the experi-mental positron annihilation characteristics may lie inthe approximations made in the calculations of the elec-tron and positron structures. For example, the LDA forthe electron exchange and correlation has been shown tooverbind atoms so that the bond lengths in molecules andsolids are too short. This deficiency may also affect therelaxation of the ions around the defects.

The positron-induced relaxation has been estimatedpreviously by Laasonen et ah. " in the case of the neutralAs vacancy in GaAs. These authors restricted the ionicrelaxation to the pure breathing-type movement of theatoms neighboring the vacancy and used the CONV

scheme in which the sum of the total electronic energyand the positron energy eigenvalue is minimized. In thismodel, the positron induces an outward relaxation of 4%of the bond length. Gilgien et al. ' considered the posi-tron state at the singly negative As vacancy in GaAs.Their calculation corresponds to the GGGC-scheme cal-culations; the most important differences are the use of adifferent pseudopotential for valence electrons and amore approximate treatment of the positron-ion core in-

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FIG. 9. Angular correlations of annihilation photons corre-sponding to the triply-negative Ga vacancy in GaAs. Themomentum distributions are integrated along the [110] direc-tion. The results are based on the self-consistent solution forthe electronic structure, positron state and the ionic positions atthe vacancy within the two-component BN (a) and GGGC (b)schemes. The experimental map by Manuel et al. (Ref. 16) isalso shown (c)~ The contour spacing is one tenth of the max-imum value. The cuts along the [100] (d) and [110](e) directionsare shown. The experimental results (Ref. 16) are given in (d)and (e) by solid lines, the results of the BN (solid circles) andGGGC (crosses) schemes are given at the discrete points forwhich the calculations are performed. The open circles refer toB¹cheme calculations in which the nearest neighbor ions ofthe vacancy have relaxed inwards 4%%uo of the bulk bond length.

10 960 M. J. PUSKA, ARI P. SEITSONEN, AND R. M. NIEMINEN 52

teraction. Gilgien et aI. found that the positron inducesa strong outward relaxation of the ions neighboring thevacancy with low-symmetry ion configurations as a re-sult. According to their calculations the positron lifetimeincreases strongly, about 50 ps. when the positron istrapped from the bulk state to the As vacancy. This isclearly more than the experimental increase of -30 ps.A more accurate treatment of the positron-ion-core in-teraction may reduce the discrepancy. Gilgien et al. alsocalculated the 2D-ACAR map for the As vacancy andfound, in agreement with our experience with the GGGCscheme for the Ga vacancy, a narrower and more isotro-pic distribution than the measured one.

IV. SUMMARY

We have developed a calculation scheme that solvesself-consistently the electronic structure, positron state,and ion positions when a positron is trapped by avacancy-type defect in a solid. The scheme is based onthe two-component density-functional theory. Thescheme is tested in the case of a triply negative Ga vacan-cy in GaAs with several descriptions for the electron-positron correlation. The calculated positron annihila-tion characteristics, i.e., the positron lifetime, the coreannihilation parameter 8' and the 2D-ACAR maps arecompared with their experimental counterparts.

The scheme proposed by Boronski and Nieminen forthe electron-positron correlation gives descriptions forthe positron potential and density that are similar tothose obtained in the conventional scheme, in which thepositron does not affect the average electron density. Thepositron state is in the BN scheme very sensitive to ionpositions, and changes in the relaxation may change thepositron annihilation characteristics remarkably. Thescheme by Gilgien et al. ' results in a more attractivepositron potential at the defect and in a more localizedpositron wave function.

We find that at a clean triply negative Ga vacancy thenearest-neighbor ions have relaxed strongly inwards. Theintroduction of the positron at the vacancy induces acompensating outward relaxation, as this reduces thepositron-ion repulsion. In the Boronski-Nieminenscheme the positron-induced relaxation is not very strongand the resulting positron lifetime is clearly shorter thanthe experimental value. However, small changes in theion relaxations can bring the theoretical and experimen-tal lifetimes into perfect agreement. This is true also forthe other annihilation characteristics. In the Gilgienet al. scheme the positron-induced relaxation is stronger;it practically heals the vacancy to the ideal one. As a re-sult the positron lifetime calculated is slightly longer thanthe experimental one. However, the 8' parameter ismuch too small and the 2D-ACAR maps too isotropicdue to the strong positron localization.

For a given ionic configuration the conventionalscheme predicts all positron annihilation characteristicsconsidered in good agreement with the two-componentBoronski-Nieminen scheme. This gives credence for theuse of the conventional scheme in model calculationsanalyzing the experimental results by assuming reason-able ionic configurations for the defects. Due to a feed-back effect the positron lifetimes from the conventionaland Gilgien et al. schemes are nearly identical, althoughthe 8'-parameters and 2D-ACAR maps differ remark-ably. This means that especially the core annihilation 8'parameter determined for the defect is a sensitive quanti-ty to describe the localization of the positron wave func-tion in addition its high value in probing the chemical en-vironments of defects in solids.

ACKNOWLEDGMENTS

The authors wish to thank R. Ambigapathy and A. A.Manuel for providing the experimental 2D-ACAR mapsfor the Ga vacancy in GaAs.

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