Positron annihilation in 3d metals

Physica 63 (1973)235-247 0 North-Hullbrsd Publishing Co.

POSITRON ANNIHILATION

IN 3d METALS

I. BAND-STRUCTURE CALCULATION OF MOMENTUM IN COPPER AND FERROMAGNETIC IRON

DENSITIES

P. E. MJJNARENDS Reactor CentrumNededand, Petten, Nederland

Received 15 May 1972

A calculation has been made of the contributions of the 3d and conduction electrons to the momentum density of photon pairs from positron annihilation in copper and ferromagnetic iron. A fast approximation scheme of Hubbard gives the band Structure and the wave functions, which are used to compute the overlap matrix element. The momentum density is computed along the , (1 lO>, and (11 l> directions in copper. A selection rule prevents large parts of the band structure from being sampled by the positron. The results are in qualitative agreement with experimentally observed features such as higher-momentum components and core anisotropy. Calculations for ferromagnetic iron, based on the lattice potentials of Wakoh and Yamashita and of Wood show that positron annihilation should enable one to distinguish between the band structures resulting from these potentials. A comparison with experiment is given in the following paper.

1. Introduction. In the past there have been various attemptsl-“) to correlate two-quantum angular correlations from the annihilation of thermalized positrons in metals with the shape of Fermi surfaces. As a rule these attempts have been most successful in simple metals, where the core contribution is small and isotropic, and where the Fermi surface does not greatly deviate from a sphere. In such metals the contributions of conduction and core electrons can be separated with a reasonable amount of confidence. Eventual discrepancies between correlation curves computed from the shape of the Fermi surface alone and the experimental curves can be removed by estimating the overlap matrix element in the independentparticle model

of an electron in the jth band with wave function y&r) = N-*u~,~ (r) exp (ik r) and a thermalized positron with wave function IV-%$ + (r), where N is the number l

235

236

of atoms and p the photon-pair

P. E. MIJNARENDS

momentum. Eq. (1) can also be written as

k being a vector of the reciprocal lattice. As knuwn, this overlap matrix element determines the contribution of an electron in the state (k, j) to the total two-photon momentum density Ki=p-

Here f(k, j) represents the Fermi-Dirac distribution function, which at T =f 0 is zero if the state (k, j) is empty, and 1 (or 2 in the case of spin degeneracy) if it is occupied. Eqs. (2) and (3) stress the fact that the annihilation of an electron of wave vector k will not only give rise to photon-pair momenta p = k (normal or N-processes), but also to pairs with p = k + Ki (umklapp or U-processes). In simple metals a few plane waves or OPW’s will usually suffice to estimate A (k, p) for the conduction band. More complicated is the situation in metals where the conduction band overlaps with other bands, e.g. the transition metals and the rare earths. It is then not sufficient to consider only the conduction band, but all bands in the neighbourhood of the Fermi surface must be taken into consideration if one attempts to interpret the observed angular correlations. The strong d character of the wave functions of some of the states at the Fermi surface, moreover, causes a strong k dependence of the overlap matrix element and makes an interpretation in terms of the s@e of the Fermi surface alone impossible. It then becomes indispensable to perform a band-structure calculation and use the wave functions to compute the matrix element. A good example of such a calculation is that of Louck@) on yttrium, which explained the hump in the angular correlation observed by Williams and Mackintosh4) in the case that the z component of the photon-pair momentum was oriented along the c axis of the crystal. In recent years several workers have studied the two-photon momentum distribution in copper with the aid of various techniques1*3*7-g). They found indications for an anisotropy of the core contribution, for the interpretation of which a knowledge of the matrix element for the 3d band is essential. A related problem is encountered in the study of the spin density in magnetized ferromagnets by means of polarized positrons1*-12). A careful analysis of the momentum distributions of the two spin populations reveals that the difference between these distributions cannot be explained from differences in Fermi-surface geometry alone, but that the matrix element is clearly spin dependent. For these reasons a numerical calculation has been made of the overlap matrix element for the 3d and conduction bands in copper and magnetized iron. Since only few calculations of this kind exist4*6*13- I5) it was decided not to attempt a

BAND-STRUCTURE

CALCULATION

OF MOMENTUM

DENSITIES

237

highly perfected computation, but rather aim at a fast calculation of limited accuracy, which would yield the main features of the momentum distribution in a semiquantitative way. For the same reason the contributions from the inner shells (3p, 3s, . . .) have not been considered. They will be relatively small and isotropic. A rapid scheme for the approximate calculation of the electronic band structure of the 3d transition metals has recently been developed by Hubbardl”). This scheme has been taken as the starting point for the calculation of the matrix element. From it the wave functions of the 3d and conduction electrons were extracted, while after some minor modifications the scheme could also be used to compute the positron wave function. The overlap matrix element was then obtained by Fourier-transforming the product of the electron and positron wave functions. Section 2 of the present paper gives a short summary of the method of computation, for a full treatment of Hubbard’s approximation method the reader is referred to the original paper. In section 3 a selection rule is derived which enables one to determine which energy bands may contribute in principle to the momentum density at a specific momentum p. The results of the numerical calculations for copper and ferromagnetic iron are presented in section 4. 2. T..eory. The approximation scheme of Hubbardl”) is based on the KorringaKohn-Rostoker-Ziman (KKRZ) formulation17) of the band-structure problem. The pseudopotential V in the original KKRZ secular equation contains the phase shifts of the various partial waves scattered off the muffin-tin potential. As the I = 2 phase shift displays a resonance in the energy range of interest, V is divided into a weak nearly-free-electron (NFE) part V’ and a d-state resonant part Vres. The secular equation may then approximately be reduced to a low-order equation of the form k2

-+V'

h"

eI

h A

-

B

&I

>(>

= 03

a

where I denotes the unit matrix. The upper left-hand corner describes the NFE bands, the lower right-hand 5 x 5 matrix the d bands, and the off-diagonal matrices h and h* cause the hybridization between these bands. The elements of the column matrix B are the coefficients in a plane-wave expansion of the pseudowave function in the outer part of the atomic polyhedron:

(5) where z represents the volume of the atomic cell. The five elements of LCcontain the resonant part of the pseudo-wave function. The NFE block in (4) consists of

238

P, E. MIJNARENDS

only 4-7 plane waves, so the size of the matrix is not more than 9 x 9 for fee metals, or 12 x 12 for bee metals. The remaining plane waves only cause small corrections in the resonant part of the matrix. This treatment of the band-structure problem is very similar to the interpolation scheme of Hodges et al. 18) and to the work of Heinelg), with the difference that the matrix elements of the model hamiltonian in (4) do not contain any adjustable parameters, but are all computed from first principles. Eq. (4) may be solved by standard techniques. Its eigenvalues E give the band structure and the corresponding eigenvectors determine the pseudo-wave function20). For r < ri, the radius of the inscribed sphere, the pseudo-wave function differs from the true-wave function. In this region the following approximation is therefore chosen

(6) where R,(r) represents the solution of the radial Schriidinger equation for the eigenvalue 8. The coefficients Xirn are determined by matching the solutions (5) and (6) at the surface of the inscribed sphere r = ri. The positron wave function was computed in an analogous way, retaining only the NFE part in (4). Apart from the sign, the positron potential was taken equal to the potential in which the electrons are moving. A correction for the absence of the exchange potential was judged unnecessary as it turned out that the functional dependence on p of the overlap matrix element Aj (k, p) is not sensitive to the form of the positron wave function. Finally the A, (k, p) were computed for a large number of k andp vectors by taking the Fourier transform (1) of the wavefunction product. With (5) and (6) for the electron and positron wave functions this yields

where the superscript + labels quantities relating to the positron, and k, = k + K,. 3. SeZection ruZe. As a result of the point symmetry of the lattice, the overlap matrix element will be zero for certain energy bands dependent on the symmetry of p. As this means that certain features of the band structure are in principle unobservable with positrons, it is desirable to express this fact in the form of a selection rule. To this end the matrix element is written in the form (2), and its

BAND-STRUCTURE

CALCULATION

OF MOMENTUM

DENSITIES

239

transformation properties under the operations of G,(k), the group of 8, are considered. As the positron is in its ground state k, = 0, its wave function ++ transforms according to the r1 representation of the cubic group and can therefore be ignored. The integral will give a nonzero result only if exp (i& r) contains a part transforming according to the same representation of G,(k) as uk. From this it is immediately seen that when & = 0, i.e., p = k, only bands corresponding to the totally symmetric representation (e.g., A,, X1, A,) will give a nonvanishing result, If Ki # 0, i.e., p on the surface of, or outside the fist Brillouin zone, other bands may contribute in addition. Operating with the projection operator21) l

on exp (i& r) will project out the sum of all parts transforming according to the jth irreducible representation of G,(k). Here h is the order of G,(k), Zjthe dimension of the jth representation, xj(R) the character of the operation R in this representation, PR the operator corresponding to R, and the summation runs over all R in G,(k). Thus, if # exp (i& r) = 0 and p = k + &, then Aj (k, p) = 0, i.e., the band corresponding to the jth representation is unobservable at this p. If, on the other hand, the projection operator yields a nonzero result, the corresponding band may contribute. Since the positron does not play a role in this argument, this selection rule is also applicable to X-ray Compton profile measurements22). l

l

TABLE I

__ Energy bands that in principle may contribute to the momentum density Q(P) for specific momenta p of annihilation radiation in fee and bee lattices with lattice constant a

(am

Observable bands

P

fee %% x 0, x, x % x, x 3-x,+,3+x

(0 < 1x1 < *)

QltQ2

x,x,1

(0< 1x1< +I

Sl,

0,

(0

LZ3

x,

1

4 &, S1, S3 A1

<

1x1 <

+I

53

bee

0, 0, x 0, x, x x, % x

Al Cl kFl33

XA3

(Ox 1x1-e

+-%3--x,++x

(0

<

1x1

<

3)

h,F2J%

(0

<

1x1

<

+I

G,

0,

+ -

x,t+x

ns3

G4

240

P. E. MIJNARENDS

Table I shows which bands may contribute to the momentum density at a few specific momenta. The observability of bands at other p’s is easily determined with the aid of (8). 4. Numerical results. 4.1. Copper. For the computations for copper the potential of Chodorow23), as given by BurdickZ4), was used. A total of 89 plane waves (7 shells) were included, while corrections were made to account for the remaining shells. The rms error in the energy, obtained by comparing 54 levels between -0.1 and + 0.9 Ry with Burdick’s values, was 0.0054 Ry. The positron energy was + 0.52 Ry with respect to the constant potential in the interstitial region. Fig. la shows the Brillouin zone and fig. 2 the contributions of the various bands for p along (loo), (1 lo), and (111) together with the band structure in these directions. Comparison of results obtained with different numbers of plane waves and at several equivalent p’s suggests that the accuracy of e(p) is of the order of 2 Ok. A slightly larger error may be caused by the neglect of all terms

Fig. 1. The Brillouin zone for (a) the fee lattice, (b) the bee lattice.

with I > 2 in the wave function inside the muffin-tin spheres, i.e., in the third term of (7). Figures about the magnitude of the latter error are not available for copper, but inclusion of the I = 3 term in the case of iron increased e(p) by about 3 % in the intermediate momentum range. The effect of hybridization is clearly visible in the contributions of the separate bands. The contribution from the lowest A1, I&, and A, bands drops off rapidly as k increases and the corresponding wave functions lose their s character owing to hybridization with a 3d band. The contribution from the second band (in the C case the third band) starts out as p2 at low momenta, as expected from a 3d band, but then rises rapidly due to the s-p character assumed by the wave function

BAND-STRUCTURE

CALCULATION

MOMENTUM

OF MOMENTUM

DENSITIES

241

p IN IURAD

Fig. 2. The calculated two-photon momentum density from the 3d and conduction bands in copper along the (NO), (1 lo), and (11 l> directions in momentum space. p is expressed in units of 1W3 mc. The thinly drawn curves represent the contributions of the separate A1, X1, S1, and Al bands (the 53 contributions are negligibly small). Broken parts of curves correspond to empty portions of partially filled bands. The symbols between parentheses refer to the band structures shown in the insets. The thinly drawn bands cannot contribute for symmetry reasons; the broken lines indicate the Fermi level.

242

P. E. MIJNARENDS

after hybridization. The net effect of hybridization on the total of all bands is practically nil, which is an expression of the NFE character of the conduction band in copper. Along (loo} the effect of the umklapp process is clearly seen This U-process has been observed experimentally’ 5). Furthermore, a comparison of the 3d-band contributions in the region 5.5 c p < 8 mrad along (100) and (110) shows that the two lowest rll bands together give a larger contribution than the lowest A, band. This agrees with the experimentally observed anisotropy of the 3d contributiong).

Fig. 3. Fermi surfaces of (a) minority- and (b)-(d) -majority-spin bands in ferromagnetic iron (after Wakoh and Yamashita, ref. 27). The surfaces marked e(h) are electron (hole) surfaces. (c) and (d) contain only hole surfaces.

4.2. Iron.. The computations for ferromagnetic iron have been performed for two crystal potentials, uiz. that of Wood26), and the self-consistent potential of Wakoh and Yamashita (WY)“), In the latter model account is taken of the ex-

BAND-STRUCTURE

CALCULATION

OF MOMENTUM

DENSITIES

243

change splitting between the majority- and minority-spin bands by employing different potentials for these bands. The potentials of Wood and WY yieldenergy bands and Fermi surfaces which are very similar throughout most of the Brillouin zone. Fig. 3 shows the Fermi surfaces resulting from the WY model. The majority (+) spin hole arms originate in the G,-branch lying above the Fermi level for all R’s on the line HNH, as is shown in fig. 4a. Thus WY find that the hole arms are continuous at N. The potential of Wood produces a different ordering of the energy levels at N, resulting in a crossing of the GZ- and G,-branches (fig. 4~). As a consequence the hole arms are pinched off at N in the model of Wood. The results of Wood are sustained by a recent calculation of Duff and Das28), which involves an accurate treatment of the exchange and correlation effects. This calculation produces the same level ordering as that of Wood, but differs from it in that the minority (-) spin hole pocket at N is found to be absent because E(N,$ < EF. In the model of WY this Docket is associated with the second NI level (fig. 4b).

N3 N4 -w Nil

0c Fig. 4. Band structure

in bee iron along HN. (a) and (b): Wakoh and Yamashita, and minority-spin bands, (c): according to Wood.

majority-

The available experimental evidence supports the gross features of the Fermi surfaces sketched in fig. 3, but does not permit definite conclusions to be drawn concerning the continuity of the hole arms. De Haas-Van Alphen experimentP) have not produced evidence for the existence of an orbit corresponding to the extremal cross section of the hole arms at N. On the other hand, in magnetoresistance measurements3*) open orbits have been observed in the (loO> and (110) directions, suggesting continuity of the hole arms or the presence of small gaps at N over which magnetic breakdown can occur. In view of the inconclusive evidence it is of interest to see whether positron annihilation can shed any light on this question. To that end the momentum

244

P. E, MIJNARENDS

density has been computed for both models following the method indicated in sections 2 and 3. The use of 87 plane waves yielded an accuracy comparable to that of the copper calculation. For the computation of the positron wave function in the model of WY the minority-spin potential was used. This potential will contain the smallest exchange contribution owing to the small number of electrons in this band.

p IN WLLIRADIANS

Fig. 5. The calculated two-photon momentum densities for the majority and minority-spin bands in ferromagnetic irun. Full curves: majority-spin band; broken curves: minority-spin band; chain curve: minority-spin band in the absence of the hole pocket at N. Curves (a)-(c) apply to the band-structure mode1 of Wakoh and Yamashita, curve (d) to the model of Wood.

Figs. 5a-5c show the momentum densities along the (loo), (1 lo), and (111) directions for the two spin populations of the WY model. The spin dependence of the momentum density is of course a consequence of the use of a spin-dependent crystal potential. In fig. 6 the contributions of the various energy bands in the part of the rHN plane inside the first Brillouin zone are shown separately. The graphs in this figure clearly illustrate the effect of the selection rule and show the elements out of which the momentum density in the two spin bands is built up. These elements are: 1) in the 1st band a large and anisotropic density centred at p = 0, stemming from the s character of the conduction-electron- wave functions; 2) in the 3rd band a density along the (110) directions that peaks around p = 4.5 and 7.5 mrad and vanishes in between at N (p = 6.0 mrad); 3) in the 5th minorityand 6th majority-spin bands a high peak of density in the neighbourhood of N caused by the strong p character of the wave function associated with the Nit-

BAND-STRUCTURE

CALCULATION

OF MOMENTUM

DEETSITIES

245

level; 4) a similar peak on PH in the 6th band. The energy of the states associated with this last peak is so far above the Fermi level, however, that it plays no role in the momentum density in iron. By drawing the Fermi surface in the same graphs it now becomes immediately evident which parts of the Fermi surface are observable with positrons. For the - spin surface this will only be the hole pocket at N, while for the + spin surface the central part of the hole arms and the electron surface around I! will be visible. These pieces of the Fermi surface give rise to the discontinuities in the momentum density curves shown in figs. 5a-5c. Along the (100) directions the + spin electron surface causes the break at p = 3.6 mrad, Nl

r;

bmnd 1

HI2

N2

Nl

bk

band 2

H12

NI

NP

&?5

w5

band 3

%

Nl

H 15

N4

bmnd 4

N4

62

band 5

N3

52

band 6

NIB

Fig. 6. Contribution of the 3d and conduction bands to the momentum density in the rHN plane in iron, calculated from the band-structure model of Wakoh and Yamashita (in arbitrary units). The bands are numbered from the bottom. Upper-left triangles correspond to the minority-spin band, lower-right triangles to the majority-spin band. Broken curves indicate the lines along which bands touch. Grey parts correspond to k states occupied by electrons,

while corresponding discontinuities due to the U-process are found at 13.4 and 20.6 mrad. Along (llO> there is a large gap at N caused by the absence of the N1, peak, which otherwise would have filled up the density minimum in the 3rd band. In the - spin band this peak lies far outside the Fermi surface. In order to show that it is the absence of this peak, and not the hole pocket in the 3rd band, that gives the main contribution to the gap, in fig. 5c the - spin momentum density is also shown for the case that the hole pocket would be filled. It is seen that although the sharp edges caused by the pocket disappear, the depth of the gap is not reduced. The gap in the + spin momentum density is caused by the combination of the central part of the hole arm in the 5th band and the electron surface in the 6th band. Along (11 l>, finally, k states on rP (N-processes) and PH (U-processes) contribute. Only along this direction the - spin electron surface centred on r is in principle observable, although the predominant d character of

246

P. E. MIJNARENDS

the wave functions makes the discontinuity very weak. At momenta around 15 mrad a U-process makes the hole pockets at H visible. The different ordering of the levels at N resulting from the use of Wood’s potential has important consequences for the momentum density along (1 lo), as may be seen from fig. 7, which shows the contributions of the 3rd, 5th, and 6th band (the other bands, being very similar to the corresponding bands of WY, are not shown). The high momentum density associated with the N1# level is now transferred to the 3rd band and hence to states below the + band Fermi level.

&I

5s -/+

spin

band 3

r;2

N,

G2

+ spin

+ spin

band 5

band 6

Fig. 7. Contribution of the 3rd minority- and majority-spin bands and the 5th and 6th majorityspin bands to the momentum density in the rHN plane, calculated from the band-structure model of Wood. The contributions of the other bands are similar to those of fig. 6.

In Wood’s model, therefore, the gap at N in the + spin momentum density vanishes, as shown in fig. Sd, while the gap in the - spin momentum density is caused solely by the presence of the hole pocket. The hole arm gives a vanishingly small contribution since now there is only very little momentum density present in the 5th band. If the hole pocket would be absent owing to a slightly increased separation between the 4s-4p and 3d bands, the momentum density would . decrease smoothly along the (110) directions without showing any gap. In the WY model, on the other hand, the gap would remain. The momentum density along the (100) and (111) directions is much the same as in the WY model and is therefore not shown separately. The conclusion of this discussion is that the momentum density of the annihilation radiation is sensitive to the differences between the band structures of Wood and WY and hence provides a possibility to distinguish experimentally between those band structures. Secondly, in the model of Wood the presence or absence of the - spin hole pockets at N should give rise to sizable effects. An experimental investigation of these points with the aid of polarized positrons will be the subject of the following paper. Acknowledgements. The major part of this work has been performed while the author was attached to the Theoretical Physics Division of the Atomic Energy

BAND-STRUCTURE

CALCULATION

OF MOMENTUM

DENSITIES

247

Research Establishment Harwell of the U.K.A.E.A. The hospitality extended to him and the permission to use the IBM 360/75 computer are gratefully acknowledged. The author wishes to thank Dr. J.Hubbard for his encouragement and continuous interest in this work.

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

Fujiwara, K. and Sueoka, O,, J. Phys. Sot. Japan 21 (1966) 1947. Donaghy, J. J. and Stewart, A.T., Phys. Rev. 164 (1967) 391 and 396. Williams, D.L., Becker, E.H., Petijevich, P. and Jones, G., Phys, Rev. Letters 20 (1968) 448. Williams, R. W. and Mackintosh, A.R,, Phys. Rev. 168 (1968) 679. Berko, S,, Cushner, S. and Erskine, J.C., Phys. Letters 27A (1968) 668. Loucks, T.L., Phys. Rev. 144 (1966) 504. Berko, S. and Plaskett, J.S., Phys. Rev. 112 (1958) 1877. Sueoka, O., J, Phys. Sot. Japan 23 (1967) 1246. Mijnarends, P.E., Phys. Rev. 178 (1969) 622. Bcrko, S. in: Positron Annihilation, A,T. Stewart and L.0, Roellig, eds., p. 61, Academic Press Inc. (New York, 1967). 11) Mijnarends, P. E. and Hijfelt, M. H. H., Proc. Int. Conf. on Magnetism, Grenoble, J. Physique 32 (1971) Cl-284. 12) Berko, S. and Mills, A.P., Proc. Int. Conf. on Magnetism, Grenoble, J. Physique 32 (1971) Cl-287. 13) Melngailis, J. and DeBenedetti, S., Phys. Rev. 145 (1966) 400. 14) Gupta, R.P. and Loucks, T.L., Phys. Rev. 176 (1968) 848. 15) Stroud, D. and Ehrenreich, H., Phys. Rev. 171 (1968) 399. X6) Hubbard, J., I. Phys. C: Solid St. Phys. 2 (1969) 1222. 17) Ziman, J.M,, Proc. Phys. Sac. 816(1965) 337. 18) Hodges, L., Ehrenreich, H. and Lang, N.D., Phys. Rev. 152 (1966) 505. 19) Heine, V,, Phys. Rev. 153 (1967) 673. 20) Hubbard, J. and Mijnarends, P.E., J. Phys. C: Solid St. Phys. 5 (1972) 2323. 21) Cornwell, J.F., Group Theory and Electronic Energy Bands in Solids, p. 57, North-Holland Publishing Co. (Amsterdam, 1969). 22) Phillips, W. C. and Weiss, R. J., Phys. Rev. 171 (1968) 790. 23) Chodorow, M.I., Ph. D. Thesis, M.I.T., 1939 (unpublished). 24) Burdick, G.A., Phys. Rev. 129 (1963) 138. 25) Cushner, S., Erskine, J.C. and Berko, S., Phys. Rev. Bl (1970) 2852. 26) Wood, J,H., Phys. Rev. 126 (1962) 517. 27) Wakoh, S. and Yamashita, J., J. Phys. Sot. Japan 21 (1966) 1712, 28) Duff, IX. and Das, T.P., Phys. Rev. B3 (1971) 192. 29) Gold, AX., J, appl. Phys. 39 (1968) 768. 30) IFin, A. and Coleman, RX., Phy$, Rev. 137 (1965) A1609.