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Puska, M. J.; Aob, M.; Brauer, G.; Korhonen, T.First-principles calculation of positron lifetimes and affinities in perfect and imperfecttransition-metal carbides and nitrides

Published in:Physical Review B

DOI:10.1103/PhysRevB.49.10947

Published: 15/04/1994

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

Please cite the original version:Puska, M. J., Aob, M., Brauer, G., & Korhonen, T. (1994). First-principles calculation of positron lifetimes andaffinities in perfect and imperfect transition-metal carbides and nitrides. Physical Review B, 49(16), 10947-10957. https://doi.org/10.1103/PhysRevB.49.10947

PHYSICAL REVIEW B VOLUME 49, NUMBER 16 15 APRIL 1994-II

First-principles calculation of positron lifetimes and afBnities in perfectand imperfect transition-metal carbides and nitrides

M. J. PuskaLaboratory of Physics, Helsinki University of Technology, SF 02-150 Espoo, Finland

M. SobInstitute of Physical Metallurgy, Academy of Sciences of the Czech Republic, Zizkova gg, CZ 616-6g Brno, Czech Repuwic

G. BrauerPositron Group of the Technical University of Dresden at Research Centre Rossendorf Inc

P. O. Box 510119, D 0181-$ Dresden, Federal Republic of Germany

T. KorhonenLaboratory of Physics, Helsinki University of Technology, SF 02150 E-spoo, Finland

(Received 28 October 1993)

First-principles electronic structure and positron-state calculations for transition-metal carbidesand nitrides are performed. Perfect NaCl structures as well as structures with metal or car-bon/mtrogen vacancies are considered. The positron affinities and lifetimes are determined. Thetrends are discussed and the results are compared with recent positron lifetime measurements forgroup-IV and -V refractory metal carbides. The present analysis suggests, contradictory to an ear-lier interpretation, that positrons are trapped and annihilated at both carbon and metal vacancies.The concentration of metal vacancies detected by positron annihilation methods is probably verylow, below the sensitivity limit of other experimental methods.

I. INTRODUCTION

Re&actory metal carbides and nitrides exhibit interest-ing physical properties, e.g. , high melting points, extremehardness, and relatively high superconducting transitiontemperatures. These make them attractive for both the-oretical investigations and technological applications.Many of their desirable characteristics are critically infiu-enced by the presence of vacancies, which occur mostly onthe nonmetal sublattice. Experimental evidence showsthat, e.g. , in the rocksalt (Bl) modification of these com-pounds, up to 30% of the lattice sites may be vacant (i.e.,

up to 60% of the nonmetal sublattice). Thus, the pres-ence of vacancies in refractory metal carbides and nitridesseems to be an intrinsic property of these compounds. '4

Positron annihilation spectroscopy (PAS) is a sensitivetechnique for studying lattice defects with electron den-sity less than average, the vacancies in particular. In puremetals PAS yields very precise values of vacancy forma-tion enthalpies, and the process of positron trapping atpoint defects in pure metallic elements is more or lesswell understood.

The behavior of point defects in alloys and intermetal-lic compounds is somewhat more complicated than inpure metals. First, several types of vacancies and inter-stitials may exist, some of them with peculiar properties.In many processes, different types of defects may com-pete, thus complicating the underlying physical mecha-nisms. In this case it is important to be able to detectthese various defect types and to resolve their role in theprocesses studied.

Due to the larger variety of defects in alloys and in-

termetallics, the interpretation of the PAS experimentaldata is more difficult than in pure metals. In most cases,a thorough theoretical analysis is indispensable in orderto determine how various types of defects are reBectedin the PAS data and to draw some quantitative conclu-sions. Although the relation of positrons and vacancies inconcentrated alloys has already been discussed in severalpublications (see, e.g. , Refs. 7—9), one usually does notperform first-principles electronic structure calculationsof positron properties in defects as, for example, in Ref.10. Nevertheless, employing such calculations is gener-ally the only way to understand the experimental PASdata in more complicated systems and to get a correctdescription of the defect properties.

In principle, vacancies may occur on both metal andnonmetal sublattices in re&actory metal carbides and ni-trides. However, most experimental results indicate thatnonmetal vacancies are the predominant defects in thesecompounds, although vacancies on both sublattices havebeen reported, especially in (presumably metastable)superstoichiometric (i.e. , nonmetal excess) compounds(see, e.g. , Ref. 11 and the references therein). Theseexperiments are usually performed by x-ray diffractionand pycnometry and their sensitivity may be estimatedas not better than +0.005 in the resulting vacancyconcentration. Thus, if the concentration of vacancieson the metallic sublattice is less than 0.005, they will notbe detected by the above-mentioned methods.

Positrons are sensitive to monovacancies in the concen-tration range 10 —10, at least in pure metals. If the

0163-1829/94/49(16)/10947(11)/$06. 00 49 10 947 1994 The American Physical Society

10 948 M. J. PUSKA, M. SOB, G. BRAUER, AND T. KORHONEN 49

concentration of vacancies is higher than about 10, allpositrons are trapped and annihilated in vacancies (sat-uration trapping). In this case, the corresponding PAScharacteristics yield information not about the electronicstructure of the bulk material, but about the features ofthe vacancies themselves.

In alloys and intermetallics the near environmentof different types of vacancies may carry different netcharges. As in the case of semiconductors more nega-tively charged defects are expected to attract positronsmore strongly. Therefore, preferential annihilation atsome kinds of defects may take place, whereas some othertypes of defects are not detected at all.

Vacancy-free refractory metal carbides and nitrides ex-hibit a very high electroaic density. The correspondingpositron lifetime may be estimated to be about 100 ps;this is lower than the values found for most transitionmetals. Recent PAS measurements of Rempel et al.yield values of 155—176 ps in refractory metal carbides,which were interpreted in terms of predominant positronannihilation with metal valence electrons, taking into ac-count the presence of carbon vacancies even in nominallystoichiometric samples. It is one of the principal goalsof this paper to find out, with the help of first-principlestheoretical analysis, which types of vacancies affect thepositrons in those materials, and to gain a deeper under-standing of the experimental results.

In addition to the refractory (group-IV and -V) metalcarbides and nitrides, which exist in the Bl structure, wehave extended our calculations along the 3d transition-metal series up to Fe in order to see the trends exhibitedby various quantities more clearly. It should be possi-ble to generalize the main conclusions drawn for the B1-structure carbides and nitrides to the other structures aswell.

Another important quantity which may be used in theanalysis of the PAS data is the positron a%nity. Ina material with carbide or nitride precipitates, the dif-ference in positron afBnity between the precipitate andthe matrix determines whether the precipitates are at-tractive or repulsive for the positron and, therefore, ifthey may be detected by PAS or not. Whether sucha case occurs or not has been discussed in neutron-irradiated pressure vessel steels, and the present resultsare highly desirable to understand the PAS data obtainedin those materials. Let us remember that until re-cently positron afBnities were calculated for elementalmetals only. Here we show that such calculations aretractable also for perfect and imperfect compounds.

The paper is organized as follows: in Sec. II the detailsof the calculation of electronic structure, positron aKn-ity, and lifetime are given. Section III describes the the-oretical results and their colnparison with available ex-perimental data. The electronic structures and positroncharacteristics are thoroughly discussed as well. Conclu-sions are presented in the closing Sec. IV.

II. COMPUTATIONAL METHOD

Our self-consistent electronic structure calculations aswell as the determination of the positron states are based

on density-functional theory within the local-density ap-proximation (LDA). ' In the LDA the effective poten-tial for the electrons is written in the form

~.&(r) = ~(r) + ~ (n(r))

where P(r) is the Coulomb potential due to the nuclei andelectron charge density and v„, is the LDA exchange-correlation potential, which depends on the electrondensity n(r). The effective potential determines, via one-electron wave functions, the electron density in turn. Theensuing problem has to be solved self-consistently. Theone-electron eigenvalues give the electronic band struc-ture, the most important parameter of which, for thepresent application, is the Fermi level.

The potential affecting the positron is constructed alsoin the LDA as

where P is the Coulomb potential as in Eq. (1), and V, „is the correlation potential describing the energy loweringdue to the electroa pileup aear the positron. The corre-lation potential V, „is treated within the LDA based onmany-body calculations for a delocalized positron in ahomogeneous electron gas.

We calculate the self-consistent electronic structures ofdifferen carbides and nitrides using the linear-muKn-tin-orbital method (LMTO) within the atomic-spheres ap-proximation (ASA).20 For a given compound, the radii ofthe metal atom and the carbon or nitrogen atom spheresare chosen to be equal. The core electron wave func-tions in the solid state are frozen to be the free-atomfunctions. The valence electron wave functions are de-termined scalar relativistically using partial waves up tol = 2 for the transition-metal spheres and up to l = 1 forthe carbon or nitrogea spheres. The spheres are arrangedinto the NaCl lattice structure and lattice constaats aredetermined by minimizing the total energy.

In the calculations for the NaCl-structure carbides aadnitrides with vacancies a supercell with eight atomic sites(Fig. 1) is used. For the lattice constants the valuesoptimized for perfect crystals are employed. The atomsneighboring the vacancies are not allowed to relax &omtheir perfect lattice positions. The positron states arecalculated also in the supercell geometry using the samenumerical methods.

In the LMTO-ASA method the potentials and energylevels are given with respect to the so-called crystal zerolevel, which is defined as the zero of the Coulomb poten-tial due to the nuclei and the electron density of the infi-nite solid. In the ASA the lattice is divided into spherescentered around nuclei. The spheres fill the whole spaceand the electron density and the potentials are approxi-mated to be spherical inside these spheres. Therefore thecalculation of the Coulomb potential and the determina-tion of the crystal zero is easy. For example, in the caseof fcc and bcc metals, for which all the spheres are identi-cal neutral % igner-Seitz spheres, the Coulomb potentialdue to every sphere vanishes just outside its surface. Theposition of the Fermi level relative to the crystal zero de-fines the electron chemical potential p . For positrons,

49 FIRST-PRINCIPLES CALCULATION OF POSITRON. . . 10 949

I. The results for TiC, VC, TiN, and VN are in goodagreement with previous LMTO-ASA calculations. Forthe carbides and also for the nitrides the lattice con-stants decrease along the Sd series. However, they in-crease sharply when going downward in the Periodic Ta-

FIG. 1. NaC1 structure with a vacancy. For a vacancy inthe metal (nonmetal) sublattice the white and black circlesdenote the metal (nonmetal) and nonmetal (metal) atoms,respectively. The supercell used in the calculations consistsof eight sites forming a regular small cube. The sites areoccupied by four black and three white circles and one site isvacant.

TABLE I. Calculated properties of the NaCl-structuretransition-metal carbides and nitrides. The perfect lattices(MX, M = transition metal, X = C or N) as well as latticescontaining metal (MsX4) and carbon or nitrogen (M4X3) va-cancies are considered. The results for bcc Fe and orthorhom-bic Fe3C with the experimental lattice constants are shown,too. For the carbides and nitrides the NaC1-structure latticeconstants a are optimized for the perfect lattices. The exper-imental lattice constants (Ref. 39) are given in parentheses;for Fe3C the lattice parameters from Ref. 34 were used. p,

and p+ denote the electron and positron chemical potentials,respectively. A~ and 7 are the positron affinity [Eq. (3)j andlifetime, respectively. The positron afBnities and lifetimes dif-fer slightly from the previously published values (Ref. 14) dueto a better self-consistency and more accurate determinationof the positron potential in the present calculations.

the position of the bottom (at k = 0) of the lowest en-

ergy band relative to the crystal zero gives the positronchemical potential p,+.

The positron affinity A+ is de6ned as the sum

&+ = P—+0+.The importance of the positron affinity is that the differ-ence of the positron energies between different materialsin contact is the difference in their positron affinities. 6

Positrons have the lowest absolute energy level in thematerial which has the lowest (negative) positron affin-ity. Moreover, the positron affinity is an important quan-tity in surface physics studied by slow-positron beamtechniques. For example, the positron affinity is relatedto the measured positronium formation energy and tothe positron work function. For the relations betweenpositron energetics and the affinity in the case of solidsurfaces, see Ref. 16.

The positron lifetime v is calculated from the electronand positron densities within the LDA. In the LDA,the total positron annihilation rate A = 1/7 reads as

drn+rFn r (4)

III. KESU'LTS AND DISCUSSION

A. Electronic structures

The calculated NaCl-structure lattice constants for thetransition-metal carbides and nitrides are given in Table

where n+(r) is the positron density and I'(n ) is thepositron annihilation rate in a homogeneous electrongas with density n . We underline that the presentLDA calculations give 6rst-principles results for both thepositron affinities and the lifetimes, i.e., there are no ad-justable parameters in our calculations.

MaterialTiCTi3C4Ti4C3VCV3C4V4C3CrCCr3C4Cr4C3MnCMn3C4Mn4C3FeCFe3C4Fe4C3ZrCZr3C4Zr4C3NbCNb3C4Nb4C3HfCHf3C4Hf4C3TaCTaaC4Ta4C3TiNTi3N4Ti4N3VNV3N4V4N3CrNCr3N4Cr4N3FeFe3C

a(a)8.060 (8.175)

7.802 (7.881)

7.641

7.544

7.415

8.880 (8.878)

8.620 (8.447)

8.690 (8.766)

8.566 (8.420)

8.143 (8.013)

7.968 (7.826)

7.830

5.406

(eV)+2.0—0.9+1.9+3.3+0.3+2.2+3.1+0.6+2.4+2.8+0.8+2.2+2.4+0.8+1.8+0.7—1.6+1.0+1.9—0.2+1.2+1.4—1.8+1 7+3.0—0.5+1.8+5.0+0.5+3.5+4.5+0.8+3.4+3.8+0.6+3.4—1.3+1.9

—6.7—5.1—3.5—6.8—5.4—3.3—6.6—5.2

302—6.5—5.1—3.0—6.2—5.1—3.7—5.9—5.4—2.3—4.8—4.8

3%1—5.9—5.1—2.S—6.3—5.6—3.8—7.4—6.0—3.5—7.1—5.6—3.5—7.0—5.5—3.1—2.5

A+ (eV)—1.8—7.6—3.2—0.2—6.6—3.2—0.1—6.0—2.9—0.4—5.7—2.9—0.5—5.4

303—3.0—7.5—4.4—Q.4—5.0—3.5—1.6—7.7—3.4+0.1—6.8—3.8+1.2—6.9—2.6+1.0—6.2—2.2+0.3—6.5—2.1—4.4—0.6

7 (ps)98

16112489

14911784

14111282

136109

7812910611419614810518113810518513799

17713010015S14394

15413889

1471239991

10 950 M. J. PUSKA, M. SOB, G. BRAUER, AND T. KORHONEN

60-Total DOS

40

I

I

I

I

i

I

I

20

060

40

Tl Cl

I

1

I

I

I

I

I

I

I

ble from the 3d series to the 4d series and then they de-crease slightly between the 4d and 5d series. These trendsreHect the changes in the localization of the uppermostd wave functions. The calculated lattice constants arecompared with the experimental values in cases wherethe NaCl structure is stable. The theoretical lattice con-stants calculated in the I DA are expected to be smallerthan the experimental ones, but in our results this is nottrue for ZrC, NbC, TaC, TiN, and VN. Moreover, the ex-perimental trend that the lattice constant for a nitride isslightly smaller than that for the corresponding carbide isnot obeyed in the calculated results (Table I). The reasonfor these small discrepancies is most probably the ASA,which cannot describe open lattice structures accuratelyenough without empty spheres. The NaCl structure ison the borderline between dense (fcc, bcc) and open (di-amond) lattice structures.

As an example of the calculated electronic structureswe show in Fig. 2 the density of states (DOS) for TiC.Again, the present results agree very well with previousKKR (Korringa-Kohn-Rostoker) 2s and LMTO-ASA24 2s

calculations. In the total DOS the 6rst structure on theleft is due to mainly 8-type states of carbon. The broadstructure just below the Fermi level corresponds to thehybridized C p and Ti d bands. Above the Fermi levelthe main contribution to the DOS comes f'rom the Tid bands. The DOS's for the other NaCl-structure car-

bides and nitrides are very similar to that for TiC. Thus,the relative position of the the Fermi level for the dif-ferent compounds depends primarily on the number ofvalence electrons. For the group-IV refractory metal car-bides TiC, ZrC, and HfC the Fermi level is near a deepminimum of the DOS, whereas for the other carbides andnitrides considered in this work the Fermi level is situatedin the high-DOS region of the metal d bands. The po-sition of the Fermi level relative to the DOS is re8ectedalso in the behavior of the electron chemical potential,i.e., in the position of the Fermi level relative to thecrystal zero. According to Table I the electron chemi-cal potential for carbides rises strongly ft.om TiC to VCand then decreases slowly toward FeC. The nitrides TiN,VN, and CrN, which we have studied, are in the regionof decreasing electron chemical potential (the number ofvalence electrons in TiN is the same as in VC). In the caseof transition-metal carbides, going down within group IVor group V first lowers the electron chemical potential be-tween the 3d and 4d series, but increases it between the4d and 5d series. This behavior reBects again changes inthe localization of the d wave functions or in the band-widths.

The eEects due to introduction of carbon or nitrogenvacancies agree with previous notions24' s' [the mainreason for the minor differences between our DOS's andthose in the above papers is probably the use of a largernumber of k points (84 in the irreducible Brillouin zone)in our calculations]. The reduction in the number ofvalence electrons has a tendency to lower the electronchemical potential. As seen in Fig. 3, in Ti4C3 the car-bon vacancies have only small effects on the positions ofthe different band edges. But the prominent features arethe induced peaks in the DOS at the minimum betweenthe carbon-p —metal-d hybridized bands and the metal dbands (directly at the Fermi energy in the case of Ti4Cs).These peaks have the metal d character and they are dueto the breaking of the carbon-metal bonds. 24 Because for

20

200-Ti4C3

020 - Tip

20 - Tis

20 - Cp

020 - Cs

I I

I

I

1

I

I

I

I I

I

I

I

1

I

I

I I

I

l

I

I

II

I

J

1

I

I I

-0.8 -0.6 -0.4 -0.2ENERGY (Ry)

0.2

FIG. 2. Total density of states for TiC and its diferentcomponents. The Fermi level is denoted by a dashed line.

100

200Ti3C4

I

I

I

I

l

I

I

I

I ~I

100

-0.8 -0.6 -0.4 -0.2ENERGY {Ry)

0.2

FIG. 3. Total density of states for the vacancy latticesTi4C3 and Ti3C4. The Fermi level is denoted by a dashedline.

49 FIRST-PRINCIPLES CALCULATION OF POSITRON. . . 10 951

the group-IV carbides the Fermi level is around the min-

imuxn in the DOS and thus at the same region as theinduced peaks, the Fermi level is pinned by these peaks.As a result, the electron chemical potential decreases onlyslightly in TiC, and in ZrC and HfC it even increasesdue to the introduction of carbon vacancies. In the casesof the other carbides and nitrides studied, the vacancy-induced peaks are at low'er energies relative to the Fermilevel and the carbon or nitrogen vacancies decrease thevalue of the electron chemical potential.

As discussed in Refs. 29 and 30, some of the nonmetalvacancy peaks may be due to the long-range ordering ofthe vacancies in the calculated structures. Therefore thenuxnber, intensity, and position of the vacancy-inducedpeaks may be diferent for disordered systems or for iso-lated vacancies. However, the position of the Fermi leveland the value of the DOS at the Fermi level will hardlyinBuence the calculated positron lifetime, as it is deter-mined by the total electron density [Eq. (4)j. Theremight be some minor changes in the calculated values ofpositron aKnities, but, as our results show, they cannotinfiuence the physical conclusions drawn in the presentpaper. Thus, as far as the calculated positron lifetimesand affinities are concerned, they may be only slightlychanged due to vacancy ordering. As we are primarilyinterested in isolated vacancies, we have made conver-gence tests of positron annihilation characteristics withrespect to the size of the elementary cell. The results ofthose tests are described in Sec. III C.

As we can see &om Fig. 3, the metal vacancies do notinduce new prominent peaks. The Fermi level is loweredwith respect to the perfect crystal both due to the overallshift of the bands and due to the reduction in the numberof valence electrons.

The above details of the electronic structures are im-portant for the positron properties as well, namely, thebehavior of the Fermi level makes a large contribution tothe trends in the positron affinity (as will be discussedbelow). Moreover, the fact that these systems are metal-lic guarantees that the position of the electron chemicalpotential is well defined, in contrast to the case of semi-conductors or insulators, in which its actual position inthe band gap is determined by the experimental condi-tions. Finally, the metallic character means that formulasfor positron correlation energy and for the annihilationrate that are based on the free-electron-gas model areexpected to be valid.

TiC

[110]

vc~ ~~ ~

[110]~ ~~ ~

the results for the positron lifetimes we have also ap-plied the superimposed-atom method for the presentsystexns. Here the electron density is non-self-consistent(it is given by the superposition of the free-atom den-

sities), but the benefit of this approach relative to theLMTO-ASA is that the real three-dimensional geoxnetryis treated correctly (the potentials and charge densitiesare not assumed spherical). The positron densities in theperfect TiC lattice as well as in the lattices Ti3C4 andTi4Cq containing Ti and C vacancies, respectively, areshown in Fig. 4. The plane of the figure is the (110)cut of the supercells used in the LMTO-ASA vacancycalculations (Fig. 1). In the perfect lattice the positronwave function has its maxixna at the interstitial regionsand vanishes toward the Ti or C nuclei. In the case ofthe Ti vacancy the positron density is clearly localizedaround the vacant site, whereas for the C vacancy thepositron exhibits a stronger diffusion toward the super-

B. Positron states~ ~~ ~

[110]

~ ~~ ~

The values of the positron chemical potential calcu-lated by the LMTO-ASA method are given in Table I.For perfect lattices the positron chemical potential variesonly slowly &om one system to another. It decreaseswhen the open vob~me experienced by the positron in-creases. The introduction of metal vacancies lowers thepositron chemical potential on the average by 3 eV,whereas the lowering by the smaller-sized carbon or ni-trogen vacancies is about 2 eV, only.

In order to visualize the positron states and to test

FIG. 4. Positron density in perfect TiC and the vacancylattices Ti4C3 and Ti3C4. The panels show the regions of the(110)planes limited by the borders of the supercells used. Foreach panel the contour spacing is one-sixth of the maximumvalue. Normalizing the positron densities so that there is onepositron inside the supercell, the maximum positron densitiesare 0.0038ao 0 0139ap, and 0.0099a~ for Ti4C4, Ti3C4,and Ti4C3, respectively. The black triangles and squares de-note the positions of the carbon and titanium nuclei, respec-tively.

10 952 M. J. PUSKA, M. SOB, G. BRAUER, AND T. KORHONEN 49

cell boundaries. This means that the eHects due to the6nite supercell size are expected to be stronger for thecarbon or nitrogen vacancies than for the metal vacan-cies. However, the localization of the positron densityin the vacancies seen in Fig. 4 means that the positronresults, for example the lifetimes obtained, may be con-sidered also as results for isolated vacancies.

C. Positron lifetimes and afHnities

The calculated positron lifetimes for the difFerenttransition-metal carbides and nitrides are given in TableI, for both undefected and defected lattices. The life-times are shown also in Fig. 5 as a function of the unitcell volume. For a given type of system, i.e., for per-fect bulk lattices and the two types of defected lattices,the positron lifetime depends rather linearly on the unitcell volume and thus on the open volume available forpositrons. However, for 4d and 5d carbides the lifetimesare at lower values than the extrapolation &om the datafor the 3d carbides would indicate. The carbon vacancylifetimes in Sd carbides are about 1.35 times longer thanthe bulk lifetimes, whereas the corresponding ratio forthe metal vacancy lifetimes is about 1.67. For the 4d and5d carbides the difference in lifetimes between the metaland carbon vacancies increases slightly. For comparison,typical vacancy-bulk lifetime ratios for metals are about1.6 and for semiconductors less than about 1.3. It is alsointeresting to note that the lifetime difFerence betweenthe two types of vacancies is smaller in nitrides than incarbides. This is in accordance with the larger number ofvalence electrons of the nitrogen atom compared to thecarbon atom.

The calculated positron lifetimes in perfect transition-metal carbides and nitrides are low, substantially lowerthan the calculated or experimental lifetimes in

200—

180-C4

160— ~ 0

~ Metalvacancies

140 — ~

120- o

~ ~100—

00

yO0

CorNvacancies

k

Bulk

80400

I

500I

600I

700 800

UNIT CELL VOLUME (a.u.)

FIG. 5. Positron lifetimes for the perfect transition-metalcarbide or nitride lattices and for the lattices with metal andcarbon or nitrogen vacancies as a function of the conventionalunit cell volume [(lattice constant) ]. The different Bd tran-sition-metal carbides and nitrides are denoted by black andopen circles, respectively. The 4d and 5d transition-metal car-bides are denoted by black squares and triangles, respectively.

the corresponding pure metals. These low values reBecttheir high atomic and electronic densities. For example,we can consider TiC as a fcc Ti metal with the octahe-dral sites occupied by carbon impurities. The lattice con-stant of 8.06ao (or the experimental one 8.175ao) is onlyslightly larger than 7.81ao which corresponds to the ex-perimental atomic volume of Ti metal (ao is the Bohr ra-dius, equal to the atomic unit). According to the atomic-superposition calculations, the increase of the Ti latticeconstant from 7.81ao to 8.06ao (8.175ao) increases thepositron lifetime from 150 ps to 164 ps (170 ps). In thefcc lattice the positron density has maxima at the octa-hedral sites. However, the introduction of the octahedralcarbon 61ls in these sites and removes the positron intothe interstitial sites of the NaCl lattice. The overlap ofelectron and positron wave functions is then much largerand, as a result, the positron lifetime in TiC reduces to98 ps.

The difFerences between the metal and carbon or ni-

trogen vacancies deserve further comment. Let us con-sider again TiC as an example; the situation in the othercarbides and nitrides is similar. Table II shows the va-lence electron and positron distributions among the dif-ferent atomic spheres in the perfect and vacancy TiClattices (the Ti and C sphere radii are equal). The par-tial positron annihilation rates with valence and coreelectrons within difFerent spheres are given, too. Thepositron distributions are normalized to unity in eachcase, whereas the sum of the valence electron contribu-tions gives the total number of valence electrons in the

MaterialTiC

TiBC4

Ti4C3

SphereTlC

CTlCETlCTl

M N N+ A (ns )4 2.922 O.ill 0.5064 5.078 0.139 0.0211 0.912 0.512 0.0003 4 459 0 116 0 0133 2.889 0.037 0.1541 5.048 0.024 0.0031 1.433 0.381 0.0003 2.767 0.103 0.4613 5.109 0.084 0.0121 2.938 0.058 0.255

A„(ns ')0.6561.3821.8840.9820.2200.2221.8590.5660.8260.345

TABLE II. Electron and positron distributions and partialpositron annihilation rates in perfect TiC and in TiC con-taining Ti (Ti3C4) or C (Ti4C3) vacancies. The distributionsand the partial annihilation rates are given in the ASA for Tiand C spheres and in the case of vacancy lattices also for the(empty) vacancy spheres (E). All the sphere radii are equal.M is the number of spheres of the given type in the super-cell. For a given sphere, N and N+ are the total valenceelectron and positron charges, respectively, and A, and A„arethe partial annihilation rates with core and valence electrons,respectively. In the case of supercells with a vacancy thereare four difFerent types of spheres (see Fig. 1). The rows forTi3C4 (Ti4C3) are ordered so that the uppermost row corre-sponds to the vacancy sphere E and the second row to thethree C (Ti) spheres neighboring the E sphere. The thirdand fourth rows correspond to the three Ti (C) spheres andone C (Ti) sphere, which are the second- and third-nearestneighbors of the E sphere, respectively.

49 FIRST-PRINCIPLES CALCULATION OF POSITRON. . . 10 953

supercell. In the perfect lattice, there is a charge trans-fer of about one electron &om a Ti sphere to a C sphere.Because of this and the smaller ionic core size of carbonthe positron resides more in the C spheres and, therefore,the annihilation takes place preferably in the C spheresrather than in the Ti spheres.

Due to the charge transfer described above, metal va-cancies may be considered as more negative environments(they are surrounded by negative C spheres) whereas car-bon vacancies are more positive. Indeed, according toTable II within the supercell for the titanium vacancythe net charge of the vacancy sphere and of six of itsneighboring C spheres is about —3.7~e~ (e is the electroncharge). Similarly, the carbon vacancy sphere and itssix neighboring Ti spheres have a net charge of about+6.0~e~. Therefore, the positron density is better local-ized inside the titanium vacancy than in the carbon va-

cancy (see Table II). This difference is clear also from thepositron densities in Fig. 4. Because of the good localiza-tion, the annihilation at the titanium vacancy happensmainly in the vacancy sphere and in the six neighboringC spheres (see Table II), whereas in the case of the car-bon vacancy there are relatively large contributions tothe annihilation rate &om the more distant spheres, too.As a consequence, the positron lifetime for the titaniumvacancy is 37 ps longer than for the carbon vacancy.

The positron affinities calculated according to Eq. (3)are given in Table I. For the perfect lattices the affinitiesare higher than those for most elemental metals. is Thisre6ects again high electronic densities in carbides andnitrides. For the materials with carbon or nitrogen va-cancies the positron affinities are about 3 eV lower thanthe affinities for perfect crystals. In the case of group-IV transition-metal carbides this lowering is smaller dueto the pinning of the Fermi level. The decrease of thepositron afBnity due to metal vacancies is stronger, onthe average about 6 eV. The high positron affinities forthe perfect carbide and nitride lattices suggest that thesematerials could be used as effective moderators in theslow-positron beam technique. The problem may, how-ever, be the difficulty of producing high-quality singlecrystals with a low vacancy concentration.

In Table I we show data for bcc Fe, too. This is be-cause the transition-metal carbides are thought to be theirradiation-induced precipitates in reactor pressure vesselsteels. Precipitates formed by neutron irradiation havealready been studied by PAS. According to Table I thecarbide precipitates in Fe could trap positrons and beobservable only if they contain metal vacancies becauseonly in that case is their affinity for the positron lower(more negative) than the iron affinity. In order to ap-proach closer to the reality in steels we have also per-formed calculations for the compound Fe3C, cementite.This is an existing carbide, in contrast to the FeC withthe NaCl structure discussed above. Fe3C has an or-thorhombic unit cell with 16 atoms. In the calculationswe have used the measured lattice constants. The bandstructure and the density of states obtained agree wellwith those calculated recently by Haglund et aL usingalso the LMTO-ASA method. The positron annihilationresults for Fe3C are shown at the bottom of Table I. It

is seen that the Fermi level in FesC is quite high relativeto bcc Fe. This fact, in addition to the increase of thepositron chemical potential, decreases the absolute valueof the positron affinity and makes positron trapping atperfect Fe3C precipitates in steels impossible. Due toa large computation cost, we have not performed calcu-lations for vacancies in Fe3C, but we expect that theirproperties, with respect to positron states, will be quali-tatively similar to those for FeC. The consequences of thepresent positron affinity data in the interpretation of thePAS results for irradiated reactor pressure vessel steelsare discussed in another context.

The positron binding energy to one isolated vacancycan be estimated from the supercell calculations as fol-lows. In principle, one should have one vacancy in thesupercell and increase the size of the supercell until theelectron and positron energy levels do not change any-more. The positron binding energy to the vacancy wouldbe the asymptotic value of the positron afBnity difFerencebetween the vacancy system and the perfect bulk mate-rial. Because the electron chemical potential approachesthe electron chemical potential for the perfect materialwhen the supercell size increases, the positron bindingenergy would be obtained also &om the positron chemi-cal potential difference. When the supercell increases thevalue of the positron chemical potential for the vacancysystem is mainly affected by the development of the elec-trostatic potential near the vacancy. The effects due tooverlapping of positron wave functions &om the neigh-boring shells vanish rapidly because of the localization ofthe positron wave function. The net effect is, however,that the difference in the positron affinity converges morerapidly than the difference in the positron chemical po-tential. This is due to the fact that the changes in elec-tron and positron chemical potentials partly cancel eachother.

We have tested these ideas by performing calculationsfor TiC with larger supercells. In the case of a C vacancythe electron chemical potential is pinned by the peaks inthe DOS and therefore its changes are small. The changesin the positron chemical potential and consequently inthe positron affinities are small as well. For example, ifthe size of the supercell is increased &om Ti4C3 to Ti8C7,the chemical potentials and the positron affinity changeless than 0.2 eV. The positron binding energies cal-culated &om the positron chemical potentials and &omthe positron afBnities (Table I) agree then with the sameaccuracy. The case of the Ti vacancy is more interest-ing. The positron binding energies calculated &om thepositron chemical potentials and &om the positron affini-ties for different supercells are shown in Table III. Thebinding energy calculated &om the affinity difference in-creases slowly as a function of the supercell size whereasthat calculated &om the chemical potential difFerence ex-hibits a much stronger dependence. The agreement be-tween the two approaches improves remarkably if the sizeof the supercell increases. Note that the distance &oma vacancy to its nearest periodic image is the same forTl3C4, Ti7C8, and Tiq5Cq6 but the number of nearest izn-

ages decreases &om 6 to 2 in the given order. This leadsto the nearly linear behavior of the binding energies seen

10 954 M. J. PUSKA, M. SOB, G. BRAUER, AND T. KORHONEN 49

TABLE III. Positron binding energy E& at the Ti vacancyin TiC estimated from calculations with diferent supercells.The binding energy is determined as the difFerence betweenthe positron chemical potentials (b,p+) or the positron affini-ties (EA+) for the supercell containing a vacancy and for theperfect bulk lattice. The energies are in eV.

SupercellTi3C4TiyC8TigsCg6Ti3gC3g

&p(&a+)2.93.94.65.3

Zh(aA+)5.85.96.06.1

TABLE IV. Positron lifetimes 7 and binding energies Egat vacancies obtained by the atomic-superposition method.

MaterialTiCTi3C4Ti4C3VCV3C4V4C3

~ (ps)10716013198147121

Ep (ev)

4.81.7

4.81.9

in Table III. For Ti3~C32 the distance between the near-est vacancies has increased by a factor of 2 and we expectthat the binding energy calculated as the afBnity differ-ence will converge by better then 0.1 eV. These resultsshow that the positron aKnity is a very useful quantity inthe determination of the positron binding energies, too.

According to the positron afBnities given in Table I,the positron binding energies to the metal vacancies arelarger than to the carbon vacancies. This is consistentwith the differences in positron localization in these va-cancies (Fig. 4 and Table II) and with the fact that thepositron lifetimes are longer at the metal vacancies thanat the carbon or nitrogen vacancies.

The results of the atomic-superposition calculations forthe positron lifetimes and binding energies to vacancies inTiC and VC are given in Table IV. Results for the othercarbides and nitrides are similar. The supercells used inthese calculations contain 8 and 7 atoms for the perfectand vacancy lattices, respectively. It can be seen that theatomic-superposition method gives slightly longer life-times for the bulk and for the carbon vacancies thanLMTO-ASA. In the case of metal vacancies, at whichthe positron wave functions are well localized, the valuesfrom atomic-superposition and LMTO-ASA calculationsare in very good agreement. The positron binding en-ergies at vacancies in the atomic-superposition methodare calculated as the difference of the positron energyeigenvalues between the perfect bulk lattice and the lat-tice containing a vacancy. The positron binding energiesgiven in Table IV can be compared with those obtainedfrom the positron amenity differences in Table I. Accord-ing to this comparison, the atomic superposition seems tounderestimate the binding energy approximately by 1 eVin most cases. The discrepancies in the lifetime and bind-ing energies arise from the different geometrical approx-imations and from the differences in the self-consistency

of the electron densities. 2 The conclusion is, however,that the atomic-superposition calculations confirm wellthe trends found in the LMTO-ASA results; the lifetimevalues are even in very good agreement. The atomic-superposition method allows us to test the effects of thesupercell size on the positron lifetimes at vacancies. Theuse of a supercell with 63 atoms gives positron lifetimesof 174 ps and 126 ps for Ti and C vacancies, respectively.The increase of the size of the supercell seems to have op-posite effects on the positron lifetime for these vacancies,compared to the LMTO-ASA values. Nevertheless, wemay summarize that there is a good overall agreementbetween both methods, and small numerical differencesdo not inBuence the physical conclusions drawn.

D. Comparison with positron lifetime measurements

TABLE V. Comparison of the theoretical and experimen-tal positron lifetimes for the NaCl-structure transition-metalcarbides. The perfect lattices (MC, M = transition metal) aswell as lattices containing metal (Msc4) and carbon (M4cs)vacancies are considered. The lifetimes calculated (Ref. 23)for the transition metals are shown for comparison. ~tg, isthe calculated lifetime and r,„p~ denotes the experimental life-time. For a given carbide, w„ is the mean value of the theoret-ical positron lifetime corresponding to the metal and carbonvacancies.

MaterialTiCTi3C4Ti4C3TlZrCZr3C4Zr4C3ZrNbcNb3C4Nb4C3NbHfcHf3C4Hf4C3HfTaCTa3C4Ta4C3Ta

&theo (ps)98161124

146b11419614815910518113812210518513714999177130117b

r„(p )s143

172

159

161

154

&expt (ps)160 (TiC)

14?176 (ZrCp. gp)

165161 (NbC)151 (Nbcp. r2)

119173 (HfC)

174155 (TaCp, gp)

116

Reference 13.Reference 23.

Let us compare the present calculated positron life-times with recent measurements of Rempel et al. forgroup-IV and group-V transition-metal carbides in Ta-ble V. The theoretical and experimental positron life-times for the corresponding transition metals are shownfor comparison, too. These theoretical values were ob-tained previously~s by the same methods as the presentcarbide results. The calculated positron lifetimes for the

49 FIRST-PRINCIPLES CALCULATION OF POSITRON. . . 10 955

transition metals correspond very well to the xneasuredvalues in most cases. This gives us confidence in the the-oretical methods, especially the LDA, for predicting thepositron lifetimes.

It may be seen Rom Table V that the experimentallifetimes are much longer than the lifetimes calculatedhere for perfect carbide lattices. It is interesting to notethat, for example, the measured lifetime of 160 ps forTiC would correspond to the TiC lattice from which allthe carbon atoxns are removed and the remaining fccTi lattice is not allowed to relax (see the discussion inSec. III C). As a matter of fact, Rempel et al.~s arrivedat a similar result using simpler arguments. However,their conclusion was that, in the refractory metal car-bides, positrons annihilate mainly with metal valenceelectrons and that the contribution of the carbon elec-trons to the annihilation is vanishingly sxnall. Accordingto the present calculations annihilation with carbon elec-trons is important (see Table II) and the experimentallifetixnes correspond rather to configurations with vacan-cies.

A more detailed comparison of the theoretical and ex-perimental results shows that, with surprisingly good ac-curacyth, e measured lifetimes correspond to the averageof the lifetimes calculated in the structures with metaland carbon vacancies (r„ in Table V). Only in the case ofTiC does the measured lifetime seem to correspond wellto the lifetime for a titanium vacancy. On the basis ofthese results, it seems that in TiC positrons annihilatemostly at the Ti vacancies, whereas in the other carbidesthe annihilation takes place at both metal and carbonvacancies.

Rempel et al. found that the positron lifetime inNbC„continually decreases from 161 ps to 151 ps whenthe carbon content is decreased &om y = 1.00 to y =0.72. They concluded that this is due to the increaseof annihilation with metal electrons. According to thepresent results and our interpretation, the decrease ofthe positron lifetime refiects the fact that the positronannihilation at metal vacancies becomes less pronouncedin the sample with a lower carbon content, in favor ofannihilation at carbon vacancies.

There is also another argument in favor of positron an-nihilation at vacancies and not in the bulk. ManuelM hasmeasured two-dimensional angular correlation positronannihilation spectra of NbC, which were fully isotropic.He interpreted his results as an indication of positrontrapping at vacancies, as no modulation due to bulk elec-tronic moment»m density was present.

Thus, on the basis of the above discussion, it seexnsthat a decisive role of vacancies in positron annihilationin refractory metal carbides is very likely. Except in thecase of TiC, both metal and carbon vacancies should beinvolved in these materials and, therefore, one would ex-pect at least two different lifetixne components. Unfor-tunately, the resolution and statistics of the xneasure-ments presented by Rempel et al. did not allow a reli-able decomposition. That is why only average positronlifetimes were given. However, these average lifetimesfully correspond to our interpretation. Future xneasure-ments with a better resolution power and higher statistics

TABLE VI. Positron lifetimes 7 and binding energies Egat carbon vacancy clusters in TiC and NbC. N is the numberof vacancies forming the vacancy cluster. The results are ob-tained by the atomic-superposition method using supercellswith 64 —N atoxns. N = oo denotes the results for the barefcc metal lattice with the lattice constant of the correspondingmetal carbide.

MaterialTiC

N01234

01234

~ (ps)107126140147153164114133145151156165

Eg (eV)

0.71.72.42.94.2

0.71.52.12.53.5

should verify or disprove it.One could also think that the experimental lifetimes

correspond to the carbon vacancies only, in which casethe too-low theoretical values would be due to the omis-sion of the lattice relaxation around the vacancy. How-ever, using the atomic-superposition xnethod, we haveestimated that, for instance, in TiC this model requiresan outward (breathing) relaxation of the Ti atoms neigh-boring the C vacancy, which is about 10% of the nearest-neighbor atom distance. Such a large relaxation is im-probable for dense carbides or nitrides. Carbon vacancyagglomerates could also possibly cause the relatively longmeasured positron lifetimes, although their existence isquite improbable in well annealed stoichiometric sam-ples. We have studied this idea by performing atomic-superposition calculations for clusters of two, three, andfour vacancies in TiC and NbC. In these calculations thesupercells are obtained &om the 64-atom bulk supercellby removing carbon atoms which are nearest neighborsto each others and keeping the metal atoms in theirideal lattice positions. The results are shown in Ta-ble VI. The increase in the positron lifetime is ratherslow. Only three-dimensional vacancy clusters of at leastfour atoms give lifetimes which are close to the measuredones. In this case the annihilation takes place mainlywith electrons from the metal atoms, and the situationresembles the above-discussed annihilation in a transi-tion metal with increased lattice constant. (From thecalculational point of view it is interesting to note thatthe positron binding energies at carbon vacancies given inTable VI for the large supercell are smaller than those forthe smaller supercell given in Table IV. This refiects the6nite positron band dispersion in the case of the smallersupercell. )

However, the possibility of carbon or nitrogen vacancyclusters must be considered as highly hypothetic in re-fractory metal carbides and nitrides. Carbon vacanciesform ordered superstructuress 4 and, therefore, their in-

10 956 M. J. PUSKA, M. SOB, G. BRAUER, AND T. KORHONEN 49

teraction is repulsive (see also Ref. 38, where the efjectivecluster interactions for carbon vacancies in substoichio-metric VC have been coinputed). Thus, the formationof three-dimensional carbon or nitrogen vacancy clustersis not favored. The nonmetal vacancies can be closer to-gether only if their concentration is high enough. How-ever, this was not the case of the samples used in Ref.13, which were nearly stoichiometric. The existence ofcarbon vacancy clusters in those samples is, therefore,highly improbable.

Thus, the comparison of our theoretical positron life-times with the measured ones strongly indicates the ex-istence of metal vacancies in the samples. Their concen-tration may, however, be small in comparison with thatof the carbon vacancies, if the positron has a much largerpreference to annihilate at the former, i.e., if the positrontrapping coefficient for the metal vacancies is much largerthan for the carbon vacancies. As a matter of fact, thisis probably the case, because the binding energies to themetal vacancies are larger than to the carbon vacanciesand the trapping coefficient (for the electron-hole excita-tion process relevant here) increases as the binding energyincreases. Moreover, it is known that positron trappingat positively charged vacancies in semiconductors practi-cally vanishes due to the repulsive Coulomb barrier, al-though there is a bound positron state at the vacancy.According to Table II, the nearest environment of thecarbon vacancy has a net positive charge whereas thatof the metal vacancy has a net negative charge. The netpositive charge may lead to a repulsive Coulomb barriernear the carbon vacancy, although due to the screeningthis barrier has, in contrast to the case of a semiconduc-tor, a finite range. Thus, the different characters of thetotal charge densities for the carbon and metal vacanciescan lead to preferential positron trapping into the metalvacancies prior to the carbon vacancies.

Independent (i.e., nonpositron) evidence for the exis-tence of metal vacancies in B1-type re&actory metal car-bides and nitrides is rare. Metal vacancies in ZrN werereported in Refs. 41, 42, and 43. Straumanis founda considerably high metal vacancy concentration also inTiN and HfN (up to 6.31%%uo in nominally stoichiomet-ric HfN). A high structural defect concentration on theTi sublattice in TiN was revealed in Refs. 44 and 45.Ehrlich44 shows very nicely the decrease of the vacancyconcentration on both metal and nonmetal sublattices inthe series TiO —TiN —TiC, the defect concentration in TiCbeing zero within the experimental error limit. Thus, inrefractory metal nitrides the existence of metal vacancieshas been independently proved; in carbides they may ex-ist as well according to these measurements, but in asmall amount. From comparison of our calculations withthe experimental results we are not able to estimatethe metal vacancy concentration in re&actory metal car-bides either. It will probably be lower than 0.005, theestimated sensitivity limit of the pycnometric and x-raydiffraction methods. ' Further investigations in this di-rection would be desirable.

The only PAS results for re&actory metal nitrides we

are aware of have been published by Brunner and Perry.They measured the positron lifetime spectra in TiN andHfN thin films. These data are hard to compare withour calculations because, in our opinion, these films andtheir interface with the substrate are not sufficiently wellcharacterized.

IV. CONCLUSIONS

We have performed first-principles electronic struc-ture and positron-state calculations for transition-metalcarbides and nitrides. The quantities studied are thepositron affinity and lifetime. For perfect carbide or ni-tride lattices the positron affinities are high and the life-times are short in comparison with the values for the cor-responding transition metals. This result reQects the veryhigh atomic and electronic density of the transition-metalcarbides and nitrides. For lattices with metal or car-bon/nitrogen vacancies the positron affinities are lowerand the positron lifetimes longer than for the perfect lat-tices. The changes are larger for lattices with metal va-cancies.

Comparison of the calculated positron lifetimes withrecent measurements for group-IV and group-V re&ac-tory metal carbides suggests that even in the highly sto-ichiometric systems positrons annihilate mainly at va-cancies. Our results indicate that positrons are trappedand annihilated also at metal vacancies, the existence ofwhich is difficult to see by other experimental methods.We suppose, on the basis of the self-consistent electronicstructures, that the positron trapping coefficient is muchlarger for the metal vacancies than for the carbon va-cancies. This makes the metal vacancies visible in PASexperiments, although their concentration will probablybe lower than 0.005.

The theoretical values of positron lifetimes for re&ac-tory metal nitrides could not be compared with measure-ments because of lack of experimental data. In this sense,our results can be considered as predictions and shouldmotivate positron annihilation experiments on these im-portant materials.

The positron affinities calculated in this work areimportant parameters if used in the interpretation ofpositron annihilation results for materials containingcarbide or nitride precipitates. A discussion aboutirradiation-induced precipitates in reactor vessel steels,which might consist of carbides and/or nitrides, is pub-lished elsewhere.

ACKNOWLEDGMENTS

The authors wish to thank Dr. A. A. Rempel for stim-ulating discussions and Dr. A. Manuel for informationabout his two-dimensional angular correlation positronannihilation measurements on NbC.

49 FIRST-PRINCIPLES CALCULATION OF POSITRON. . . 10 957

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