Pseudopotential linear response method for core hole screening in metals

Solid State Communications, Vol. 56, No. 12, pp. 1073-1076, 1985. Printed in Great Britain.

0038-1098/85 $3.00 + .00 Pergamon Press Ltd.

PSEUDOPOTENTIAL LINEAR RESPONSE METHOD FOR CORE HOLE SCREENING IN METALS W. Lambrecht* Departamento de Fisica, Universidad Nacional del Sur, RA-8000 Bahia Blanca, Argentina and N.J. Castellani and D.B. Leroy Planta Piloto de Ingenieria Quimica (UNS-C.O.N.I.C.E.T.), RA-8000 Bahia Blanca, Argentina

(Received 8 August 1985 by M. Cardona) The screening energy of core holes in metals is calculated by means of linear response theory of the homogeneous electron gas, combined with a pseudopotential description of the core hole perturbation, which is shown to give results in agreement with SCF-calculations in the excited atom model. THE CORE HOLE RELAXATION or screening energy, relevant for XPS (X-ray photo electron spectroscopy), has been calculated by a variety of methods. For a recent review, see Bechstedt [1] and references cited therein. Though one of the most attractive methods is linear response dielectric theory, because it is easily applicable to different types of solids and situations, e.g. metals [ 2 - 4 ] , semiconductors [5] and ionic solids [ 6 - 8 ] , chemisorption [3], small particles [9], this method has to our knowledge not previously been shown to be in agreement with more involved ASCF density functional calculations. The excited atom model of Williams and Lang [10], on the other hand was :shown to be in fair agreement with explicit jellium calculations [11]. Although the excited atom model :is extremely simple and useful for metals, it has obvious shortcomings. It makes the ad hoc assumption that the screening charge density corresponds to adding a neutralizing valence electron to the core-excited atom, thereby making the screening energy completely independent of the host and excluding a lot of interesting situations from further study: e.g. alloys, small particles, surfaces. In this contribution, we show that the linear response theory of the homogeneous electron gas yields results in agreement with the excited atom model for simple metals, provided that the core hole perturbation is described by a suitable pseudopotential. It was already pointed out by Gadzuk [3], that the essential reason tbr the overscreening of the core hole Coulomb potential in the linear response method as applied to the homogeneous electron gas, is the lack of core-orthogonality of * Present address: Max-Planck-Institut f'tir Festk6rperforschung, D-7000 Stuttgart-80, Fed. Rep. Germany.

the screening electron wavefunctions. This idea is here explored further and turned into a practical method for relaxation energy calculations. The present method has already been used in a study of core hole screening in small metal particles [9], but the formalism is somewhat simpler in the bulk case, to be presented here. We will consider the core hole as a static perturbation, introducing essentially a defect problem of a Z*-atom, i.e. an atomic core of atom number Z with an internal core hole in the host metal Z. As a justification for the use of a homogeneous electron gas approximation, we adopt a pseudopotential description of the metallic valence electrons. In this way, the core-orthogonality is avoided, but included implicitly in the use of a soft pseudopotential, from which the core orthogonalities are projected out. The defect perturbation, introduced by the core hole is thus given by:

Vh = Vps(Z*)- Vps(Z),

(1)

where Vps is the ionic, i.e. unscreened pseudopotential of the core excited and normal ionic core respectively. As the screening energy is an average quantity, it will not depend very critically on the details of the potential away from the core excited atom. This is also confirmed by the success of the jellium [ 11 ] or spherically averaged solid models [12]. In fact, only very recently [13], defect calculations, including the actual metal structure have been performed, for core holes. Therefore, we will assume the pseudopotential Vps(Z) to be exactly flat, except at the core-excited atom, whereby the electron gas of the unperturbed problem becomes the homogeneous electron gas. In other words, the problem of a defect potential Vh in a smoothly varying pseudopotential of the actual metal is replaced by that of the same perturbation Vh in a constant potential, which may be

1073

CORE HOLE SCREENING IN METALS

1074

considered to be a zero'th order approximation in a nearly free electron picture. Once this step is introduced, the problem is reduced to standard linear response theory of a perturbation in the homogeneous electron gas. In principle, the ionic pseudopotentials in equation (1) can be obtained in the same way as normal pseudopotentials, e.g. by means of the norm-conserving pseudopotential method [14, 15]. In this paper, a further simplification is made, guided by considering published pseudopotentials [15] of atoms Z and Z + 1 in the spirit of the equivalent cores approximation [16]. The difference potential Vh was found to be fairly well described for the present purpose, by an Ashcroft type pseudopotential: = ----

r > r e,

r

(2) 1 r

~

rc,

re

where the cut-off radius r c is chosen so as to contain 95% of the core charge. Outside the core region, this potential has the expected attractive Coulomb character of a spherically averaged core hole, while the constant cut-off at r e will prevent excessive charge accumulation close to the core hole, which is precisely the netto effect of the core-orthogonality on the screening electron distribution, that we wish to include. The rest of the calculation is straightforward. The total metallic valence electron contribution to the relaxation energy (defined to be positive), is given by: R m

=

__ 1

f Vh(r)ns(r)d3r,

(3)

where the induced screening charge density ns is given by its Fourrier transform: q2 n s ( q ) = ~ (e(q) -1 -- 1)Vh(q), (4)

With the pseudopotential, given above in equation (2), and a dielectric function of the form e ( q ) = 1 + k2/q ~, the integrals can be done analytically and yield the simple result: 1 R m = 4kr2c-- ( 2 k r c -

R m _

1

16~r3 .

I" q4(1

e(q)-l)Vh(q)2dq'

(5)

O

with: ~ Vh(r) sin ( q r ) r d r .

Vh(q) = q 0

(6)

1 + e -2kre),

(7)

which, in the limit r c -+ O, reduces to the well-known point charge result: ½k. In the Thomas-Fermi approximation k is given by k 2 = 4 k F / = 2.4435 rs 1 and more generally it may be considered to define an effective inverse screening length. We have also used the SCF Bardeen-Lindhard [17, 18] dielectric function with and without corrections for exchange and correlation as introduced by Overhauser [ 19]. In Table 1, the results of this calculation are shown for core holes in simple metals and are compared to the excited atom model, as applied by Williams and Lang [10], here calculated in the Xa-approximation and spin independent. In order to make a comparison possible, it should be taken into account, that the excited atom model gives the extra-atomic relaxation energy R ea, instead of the extra-core contribution, as calculated presently. The screening energy coming from tile atomic valence electrons, given by the difference between the relaxation energy in the neutral atom and in the ion, where all valence electrons are removed should therefore be added to Rea: in short: R m = R e a + R a - - R i. The atomic and ionic relaxation energies are readily Obtained from atomic ASCF-calculations, and can also be obtained in the local density functional theory, by a suitable definition of the frozen orbital binding energy [20]. It is also of interest to consider the screening charge density. In the case of the Ashcroft type pseudopotential and the Thomas-Fermi dielectric function, the Fourier transform of equation (4) gives: =

k 4n

ns(r)

with e ( q ) the static wavevector dependent dielectric function, and atomic units are used throughout. As the core hole potential is assumed to be spherically symmetric, the screening charge density and its Fourier transform are also spherically symmetric and the Fourier expression equivalent to equation (3) is:

Vol. 56, No. 12

k

e -kr sinh(krc) r

rc

r > r c, '

sinh(kr) e -/~rc

:

(8) ,

41r

r

r

~

re,

rc

while with the other dielectric functions, it is easily calculated with a fast Fourier transform. The screening charge density for a Na core hole is shown in Fig. l, as obtained with tile various dielectric functions. Also shown is the excited atom model approximation for the extra-atomic screening charge density: i.e. the charge density corresponding to a Na 3s orbital. One observes the Friedel-oscillations in the case of the SCF dielectric functions, and the nodal structure in the core region in the excited atom model. It is important to notice that the screening energy is fairly independent of the details of the screening charge

Vol. 56, No. 12

CORE HOLE SCREENING IN METALS

Table 1. Metallic valence electron screening energy R m o f core holes is simple metals (in e V )

"~ o 4 0

I

1075

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

o >.

Metal Na Mg Al (a) (b) (c) (d)

re (a.u.) 1.46 1.38 1.20

Pseudopotential (a) (b) (c)

(d)

5.7 6.4 7.3

5.2 6.8 7.4

5.2 6.1 7.1

5.5 6.4 7.4

0.30

Excited atom

Thomas-Fermi dielectric function. Bardeen-IAndhard dielectric function [17, 18]. Overhauser dielectric function [19]. Spin-independent X a excited atom calculation of extra-atomic relaxation plus atomic valence relaxation.

density, because it only depends on an integral over it, as is clear from equation (3). Thus, although the present pseudopotential approach fails to describe the atomic character of the screening charge density inside the core region, contrary to the more involved jellium models, because we deliberately projected out the core-orthogonality oscillations, it does reproduce the screening energy very well, because the screening charge density obtained has the correct average spatial extent and integrates correctly to one electron charge, as is required by charge neutrality or the Friedel sum-rule. In the present theory, it is also clear why the screening energy depends little or not on the core hole involved, contrary to what might be expected from a Coulomb model where a more localized core hole should give a higher screening energy, in disagreement with the excited atom model or jellium calculations. Indeed, the screening electrons are prevented from entering tile core region as a whole and thus there is little difference between different internal core holes, as long as they remain well localized inside the core. The small variation of core hole screening with the spatial extent of the core hole has previously been interpreted by Matthew and Perkins [4] as being due to intra-core relaxation, which would tend to spread the core hole charge density over the whole core region. In principle this intra-core effect is implicitly taken into acount in equation (1), as well as any non-linear effects inside the core. Although the method, as presented here applies only to simple metals, it may semi-empirically be extended to transition-metals by simulating the more localized character of the delectrons, which are mainly responsible for the screening in this case, by only including part of the d-electrons in the electron gas and keeping the rest fixed to the core. Also, the valence d-electrons should only be made orthogonal to the lower d-electrons and the core-radius should thus be chosen accordingly. For 3d-metals, a point charge potential would apply. This approach was successfully used in a previous study of screening in small

Z l.u D

LU 0.20 ~,D < q,-

o o

0.10

Z Z LLI

"'t 0

m

I

-0.10

o

5

10

15

20

25

R(o.u.)

Fig. 1. Screening charge density: 4nns(r)r 2 for core hole in Na. o Thomas-Fermi dielectric function. ~ BardeenLindhard dielectric function [17, 18]. + Overhauser dielectric function [19]. ~Excited atom model: Na 3s atomic orbital charge density, calculated in X a approximation. The latter gives only the extra-atomic contribution, not the total valence contribution. metal particles, to which we refer for further details [9]. Summarizing, we have shown that linear response theory of the electron gas can be used to calculate relaxation energies in metals in agreement with SCF calculations, provided the core hole is described by a pseudopotential, whereby the core-orthogonality effect of keeping the screening electrons out of the core region is implicitly included and the homogeneous electron gas approximation justified. This method is of comparable simplicity as the excited atom model but should be applicable to a much wider class of problems, because the screening properties of the environment of the coreexcited ioncore enter the theory through the dielectric function. It has already been applied to screening in small particles and applications to screening in alloys and in semiconductors are presently under study. Acknowledgements - One of us (N.J.C.) wishes to thank the CONICET for financial support, and W.L. is indebted to the Max-Planck Geselschaft for a postdoctoral fellowship.

REFERENCES 1. 2. 3. 4. 5. 6.

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7. 8. 9. 10. 11. 12.

CORE HOLE SCREENING IN METALS 34,485 (1938). E. Louis, Phys. Rev. B20, 2537 (1979). F. Bechstedt, Phys. Status Solidi (b), 91, 167 (1979). N.J. Castellani, D.B. Leroy & W. Lambrecht, Chem. Phys. 95,459 (1985). A.R. Williams & N.D. Lang, Phys. Rev. Lett. 40, 954 (1978). R.M. Nieminen & M.J. Puska, Phys. Rev. B25, 67 (1982). C.O. Almbladth & U. vonBarth, Phys. Rev. B13, 3307 (1976).

13. 14. 15. 16. 17. 18. 19. 20.

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R. Zeller, private communication. D.R. Hamann, M. Schlfiter & C. Chiang, Phys. Rev. Lett. 43, 1494 (1979). G.B. Bachelet, D.R. Hamann & M. Schliiter, Phys. Rev. B26, 4199 (1982). D.A. Shirley, Chem. Phys. Lett. 16,220 (1972). J. Bardeen, Phys. Rev. 52,688(1937). J. Lindhard, Kgl. Danske, Mat. Fys. Medd. 28, 8 (1954). A.W. Overhauser,Phys. Rev. B3, 1888 (1971). W. Lambrecht, N.J. Castellani & D.B. Leroy, J. Electron Spec. Related Phenom., in press.