Self-consistent electron densities in metal vacancies

Solid State Communications,Vol. 16, pp. 831—834, 1975.

Pergamon Press.

Printed in Great Britain

SELF-CONSISTENTELECTRON DENSITIES IN METAL VACANCIES R. Nieminen,* M, Manninen, P. Hautojärvi and J. Arponen Department of Technical Physics, Helsinki University of Technology, SF.02 150 Otaniemi, Finland (Received 10 December 1974 by 0. V. Lounasmaa)

The Hartree—Fock method in the density functional formalism is applied to calculate the self-consistent electron densities in metal vacancies. Comparison shows that the mere Thomas—Fermi method describes fairly well the electron density over the whole metallic range. The calculated annihilation characteristics of positrons trapped at vacancies agree well with experimental results. The formation energies of vacancies are found to be sensitive to the lattice model.

highly repulsive core part of the potential, to far too

THE ELECTRON density distributions about vacancies and their aggregates in solids are ofinterest in many ways. Besides the pair-potential theory, they are the key to understanding the formation, migration and interaction energies of defects. On the experimental side, the technique of positron annihilation seems to yield direct information on the electronic structure of the interior of a lattice defect through the lifetime and the 27 angular correlation (momentum distribution) measurements.1 While positrons from an external source become thermalized and diffuse in matter, a fraction of them get trapped and annihilate at lattice

sinai] electron densities inside the vacancy. Our approach to the vacancy problem parallels that used by Lang and Kohn3 in calculating the electron densities at metal surfaces and the associated surface energies. We use the density-functional formalism of Hohenberg, Kohn, and Sham.4’5 The total energy of the electronic system is expressed as a functional of the density, i.e. (in atomic units) E[nJ

defects, thus reflecting the properties of the local electronic environment. Most of the previous efforts to calculate the displaced charge round a vacancy have concentrated on the use of scattering theory; the conduction electrons are allowed to scatter off a chosen repulsive defect potential, whose parameters are adjusted in order to satisfy the Friedel sum rule. Stott et aL2 took this procedure a step further by calculating the scattering self-consistently in the Hartree sense. However, even

J

V(r)n(r) dr +

II

n(r)n(r’) dr dr’ 2J J Ir—r~ (1) + T[nl + ~ [nJ, -~-

where V(r) is an external potential and T and ~ are functionals describing the kinetic and exchangecorrelation energies of the electron fluid. In the jellium model we do not take into account the lattice relaxation, which is likely to be quite small in most cases.6 A monovacancy is described as a hole in the uniform background with the ion density n~(r)= ne(r R 1), (2) —

their model includes the choice of the defect potential, for which they took the negative of the point ion field for the missing atom. This leads, we believe, due to the *

=

where the vacancy radius R3 is chosen to satisfy 4irR~ñ/3= Z. Here ñ is the jellium number density and Z the valency of the metal. This charge distribution gives rise to the external potential V(r) in (1). Defining an effective one-particle potential

Present address: Cavendish Laboratory, Cambridge CB3 OHE, England. 831

832

SELF-CONSISTENT ELECTRON DENSITIES IN METAL VACANCIES

Vefftflrl

~(r)+-~~~ ~5n(r) ,

(3) ~

to

—

AtI

I

These are the Hartree—Fock (HF) equations which must be solved self-consistently together with require-

8ACXGROUND

.

z w

“I

.

TF~

~~~~/IFDW

W

=

E I~i~(r)I2.

(5)

The overall charge balance requires that 0(r) ~ 0, when r -÷ Hence, the effective potential Vest approaches the value [6E~~/6n(r)1~.~ ~ The bulk chemical potential is Li ~ + .i~~(n), kF being the Fermi momentum. For E~~(n) we have used the Kohn—Sham exchange together with the Wigner interpolation formula.3 The set of equations (3)—(5) has been solved self-consistently in spherical geometry following an iterative numerical procedure to be described further elsewhere.7 It was found that the phase shifts were significantly different from zero up °°.

the to the self-consistency angular momentum achieved I =in7.the In successive electron density iterations was better than 0.1 percent.

C

I

I

1

2 DISTANCE

,~

0

~

I

~

~

I

I

IC

ment n(r)

_~

/ ~~;;.‘

1

~0.5

N

1

_________________

POSITIVE

C

where 0 is the total electrostatic potential in the systern, one can cast the problem of finding the density of N electrons into a set of one-particle equations 5 (~v2+ VeuEn;r])~i(r)= e~~~(r). (4)

Vol. 16, No.7

—

/

0.995 ~TF 5 6

r

I 1.

(a.u.)

7 8 (au.) I

I

5

6

9

7

FiG. 1. Conduction electron density in the jeilium vacancy of aluminium (r~= 2.07, Z 3). The solid curve is the self-consistent Hartree—Fock result. The Friedel oscillations at large r are presented in a magnified scale. The dashed curve is the Thomas—Fermi density and the dotted curve the Thomas—Fermi— Dirac—Weizsäcker density. ___________________________________________ I

NC

I ________

BACKGROUND POSITIVE

C >I-

-

~

7

________

~HF

ci.00~

~

~

‘IF

For comparison, we have also solved for the elec-

asa

~--

6 tron density by using the variational Thomas—Fermi (TF) method and some of its extensions. In the TF approximation, one includes in E ~equation (1)] only the electrostatic terms and the kinetic energy through a local description (see reference 8 for details). If exchange is also accounted for by a local formula one arrives at the Thomas—Fermi—Dirac (TFD) equation. The inhomogeneity corrections to the TFD method can be expanded in terms of density gradients. For a slowly varying density, the first correction 4 to the kinetic energy is of the Weizsäcker form Lw ~ dr (6)

J

~

which leads to the nonlinear Thomas—Fermi—Dirac— Weizsäcker (TFDW) integro—differential equation. The results for the electron density are shown in Figs. I and 2 for two representative metals, aluminium (the electron density parameter r 3 = 2.07, Z 3) and sodium (r~= 3.93, Z = 1). The self-consistent

8

c

0

I

1

2

I 3 L DISTANCE

10

12

r (aul

TFDW

r

I 5 (au)

I 6

I 7

8

8

FIG. 2. Conduction electron density in the jellium vacancy of sodium (r,, = 3.93, Z = 1). Theresult. solid The curve is the self-consistent Hartree—Fock Friedel oscillations at large r are presented in a magnified scale. The dashed curve is the Thomas—Fermi density and the dotted curve the Thomas—Fermi— Dirac—Weizsacker density. distribution exhibits the familiar Friedel oscillations, which are presented in a magnified scale for large r in Figs. 1 and 2. For a high-density metal like Al the TF method already gives fairly good electron densities. Surprisingly, in the case of Na the differences between the HF and the TF calculations are not larger on average. The TFDW density is better than the TF density in the case of Al but worse in the case of Na, since the Dirac term overestimates the exchange effect at low electron densities, which is a well known

Vol. 16, No. 7

SELF-CONSISTENT ELECTRON DENSITIES IN METAL VACANCIES

833

Table 1. Vacancy-formation energies in different lattice models calculated from Hartree—Fock electron densities. The numerical uncertainty is about 2 per cent. The experimental values have been obtained from reference 12 Metal

Z

r8

Jellium (eV)

Point ion (eV)

Pseudo ion (eV)

Experimental (eV)

Na Li Mg Al

1 1 2 3

3.93 3.25 2.65 2.07

0.50 0.49 0.43

0.78 1.07 1.57 3.61

0.71 0.94 1.39 2.24

0.42 0.34 0.89 0.75

—0.54

9 As a summary, phenomenon atomic we found thatinthe merecalculations. TF method describes fairly well the electron density distribution over the whole metallic range, whereas the refinements, i.e. the Dirac and the gradient terms, either individually or together, do not significantly improve the results even at higher electron densities. The negative electrostatic potential 0 due to the redistribution of electrons at the vacancy and the lack of the positive ion cause the trapping of the positron. The lifetime of the trapped positron depends directly on the electron density inside the vacancy.8 By using the self-consistent electron density the calculation of the binding energy. and the lifetime yields the values of 1.8 eV and 250 psec, respectively. The latter agrees closely with the experimental value of 246 ±4psec.’° Due to a stronger electrostatic potential 0 the TF density results to a higher binding energy of 2.6 eV whereas the lifetime value of 251 psec hardly deviates from that based on the HF density. The total momentum density p(p) for the annihilating electron—positron pair is given in the HF approximation by the formula1 p(p)

N

=

C ~ LI dr e~~t’+(r)~’

2 (7)

1(r)I where Li~is the positron wave function and C a constant factor. The distribution of the ps-component is obtained by integrating p(p) over p,~,and p,,. The curve calculated for the positron trapped at the aluminium vacancy is plotted in Fig. 3, together with the experimental points11 and the free electron parabola, which describes quite well the annihilations in bulk Al. Besides the fairly good agreement with experiment, we point out that the curve calculated here from the proper single-electron states is also close to the angular correlation curve calculated directly from the total electron density by using the mixed-density approximation for p(p).8

F 1

I

I

I

I

I

I

S

I

—

~ -

-

AL .

-

0.5

-

-

-

-

.

I

0

I

I

I

\ \~ \~

ANGLE 9 IN MILLIRADIANS

-

~

10

FIG. 3. The 27 angular correlation curve for positron annihilation in the aluminium vacancy calculated from one-electron Hartree—Fock states (solid line). The expenrnental pomts are those of Kusmiss and Stewart’1 and the dotted line is the free-electron parabola. All the curves have been normalized to equal areas.

The vacancy-formation energies, when approached from the electron theory point of view, are quite sensitive to the self-energy of the displaced charge.’~ This has led us to try, as another application, to calculate the vacancy-formation energies of simple metals using the self-consistent electron densities. The contributions to the vacancy-formation energy originate from the changes in the total kinetic, exchangecorrelation and electrostatic energies between a per. fectlattice and a lattice with one atomic cell removed from the bulk and placed on the surface. The sum of the first two contributions can be written, ignoring any Bloch wave corrections, in terms of the density n(r), the electrostatic potential 0(r) and the phase shifts 8,(k).7 The electrostatic energy difference can be calculated in a simple way for a jellium. The results

834

SELF-CONSISTENT ELECTRON DENSITIES IN METAL VACANCIES

for the vacancy-formation energies in some metals are given in Table I, together with the experimental results)~Not surprisingly, one can see the tendency towards better agreement for monovalent metals, whereas for high Z and small r, even a negative value is obtained in the case of Al. This is similar to the trend observed by Lang and Kohn3 in surface energies. We have also tried to go beyond the jellium approximation in calculating the electrostatic energy difference. By using the electron densitiesfrom the uniform background model, we have obtained the corrections by carrying out the relevant lattice sums7 for (i) pointion lattices and (u) pseudo-ton lattices. In the latter case, the bare electron-ion pseudopotential was the Ashcroft empty-core modelY~The results are included

Vol. 16, No.7

in Table 1, and it is clearly seen how the lattice corrections bring all the calculated values positive but too large and this effect increases progressively with ionic charge. The results are only slightly improved by taking into account the pseudopotential correction. We

believe that the high values for the formation energy are due to the use of a smooth electron density which ignores the screening of the ionic charges. The details and more results of the vacancy formation energy studies will be published elsewhere.7 Acknowledgements to Professor P. Jauho for his interestWeinare thismdebted work. One of us (R.N.) wishes to acknowledge the fmancial support from Osk Huttunen Foundation. —

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WEST R.N.,Adv. Phys. 22, 264 (1973); DEKHTYAR I.Ya.,Phys. Reports 9C, 243 (1974).

2.

STOTT M.J., BARANOVSKY S. and MARCH N.H.,Proc. R. Soc. A316, 201 (1970).

3. 4.

LANG N.D. and KOHN W.,Phys. Rev. B!, 4555 (1970).

5.

6.

KOHNW.andSHAM LJ.,Phys.Rev. 140, A1133 (1965). SINGHAL S.P.,Phys. Rev. B8, 3641 (1973).

7. 8.

NIEMINEN R., MANNINEN M., HAUTOJARVI P. and ARPONEN J. (to be published). ARPONEN J., HAUTOJARVI P., NIEMINEN R. and PAJANNE E.,J. Phys. F3, 2092 (1973).

9.

GOMBAS P.,Die sratistische Theorie des Atoms und ihre Anwendungen, Springer-Verlag, Wien (1949).

HOHENBERG P. and KOHN W.,Phys. Rev. 136, B864 (1964).

10.

COTTERILL R.M.J., PETERSEN K., TRUMPY G. and TRAFF J., J. Phys. F 2,459 (1972).

II.

KUSMISS J.H. and STEWART A.T.,Adv. Phys. 16, 471 (1967).

12.

MARCH N.H, J. Phys. F 3, 233 (1973).

13.

ASHCROFT N.W.,Phys. Lett. 23A, 48 (1966).