New dynamical effects in the spectra of core holes

Solid State Communications,Vol. 19, pp. 1 6 5 - 1 7 0 , 1976.

Pergamon Press.

Printed in Great Britain

NEW DYNAMICAL EFFECTS IN THE SPECTRA OF CORE HOLES G. Wendin, M. Ohno and S. Lundqvist Institute of Theoretical Physics, Chalmers University of Technology, Fack, S-402 20 G6teborg 5, Sweden (Received 12 January 1976 by L. Hedin)

Fluctuations of a core hole leads to new effects in core hole spectroscopy. They contribute to the relaxation and bring in new components in the satellite spectrum. In certain cases the fluctuations may lead to a rapid decay of the core hole. 1. INTRODUCTION WHEN a photoelectron leaves an atom bound in a solid or a molecule, the electrons in the neighbourhood will relax around the localized hole charge which has been created. The understanding of the relaxation is of fundamental importance for the interpretation of photoelectron spectra or any other deep hole spectroscopy. The conventional picture of the relaxation mechanism is that the hole acts like a fixed charge distribution and that the associated electrostatic forces tend to polarize the surrounding electron cloud. This gives rise to a relaxation shift of the.core level to the position er. The energy corresponding to a removal without allowing the other electrons to relax is denoted by Cfrozen a n d corresponds to the energy given by Koopmans theorem 1 and the relaxation shift is therefore given by e r -- 6frozen. One direct method to obtain the relaxation shift is to compare numerically the results of two self-consistent Hartree-Fock calculations, one with and one without the hole. Such calculations were done for core levels in atoms by e.g. Sureau and Berthier, 2 Bagus a and Lindgren. 4 To proceed in a more general way one needs a form of systematic many-body theory. Hedin and Johansson 5 started from the non-local and energydependent self-energy operator and derived a formula for the electronic relaxation energy which has been widely used in applications. The so called Xc~ scheme 6 takes the relaxation energy partly into account by means of the so called "transition state", which is a state in which one half electron has been removed from the orbital under consideration. An alternative way of treating relaxation is provided by the transition operator scheme developed by Goscinski et al. 7 These methods and related ones have been applied to a large number of cases and we refer to recent reviews by Basch, 8 Gelius, 9 Siegbahn a° and Shirley. n The common feature of the approximation schemes just referred to is that they all involve a mean field assumption. Although this assumption probably has a fair range o f validity we wish to draw the attention to 165

the limitations of such schemes. In particular we wish to show that there are cases where deviations from mean field behaviour are quite important and where fluctuations of the hole charge are large and will not only give large additional contributions to the relaxation shifts but will even change the qualitative features of the spectrum. In certain cases the coupling between the fluctuating hole and other fluctuations of the system lead to such a fast decay of the hole that it is not meaningful to talk about a quasi-particle state corresponding to the initial hole. In order to discuss the dynamical effects in the simplest terms possible, we choose to consider the case of electron spectroscopy in the limit of high electron velocity, corresponding to the sudden approximation. As is well known, the physics of the process can then be discussed in terms of the core hole spectrum, which will be discussed in the following section. 2. FLUCTUATIONS OF THE HOLE In the limit of high velocity of the ejected photoelectron one can consider the process as an instantaneous creation of a hole in the core o f a particular atom in the system. The energy spectrum seen is that of the N-1 electron system, which we simply refer to as the "hole spectrum". This assumes that the electron suffers no energy losses in its path from the atom to the detector, and the question of extrinsic energy losses is of no particular concern for the present study. The basic mathematical quantity to discuss in the photoemission process is the one-electron spectral weight function which is related to the one-electron Green function through the formula A(x, x'; w) = _1 [lm G(x, x'; w)l. 7r

(1)

When the theory is applied to a particular system it is often convenient to represent the spectral function as a matrix, choosing as a basis a suitable set of one-electron orbitals un(x), thus

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NEW DYNAMICAL EFFECTS IN THE SPECTRA OF CORE HOLES

Vol. 19, No. 2

Ann'(W) = f d3x dax ' u*(x)A(x, x'; w)un'(x').

(2) The energy distribution of the photo-electrons is found to be (see e.g. reference 12)

l(e) "-" e u2 ~_, Ann'(e -- w)P~npk n' nn

(3)

t

where e is the energy of the outgoing electron in state k and Pan is the one-electron momentum matrix element between the states n and k. w is the energy of the absorbed photon. Since the effects we wish to demonstrate are large qualitative effects we feel justified to use the simplest form of the theory and shall return to a more detailed discussion in a forthcoming paper. We therefore consider just the contribution from one core level and neglect non-diagonal terms. This gives the simple formula

I(e) ~ e '/2 p2nA nn(e -- w)

(4)

showing that the spectrum is directly related to core hole spectral function. Denoting the diagonal part of the core hole self energy by Enn(E), where E = e -- w, one has the explicit formula I Ann(E)

=

rr

P/~ q~

_ ~ ~ P(~q

[Im Znn(E)[ [E -- En° -- Re Znn(E)] : + [Im Znn(E)] 2" (5)

Here E ° represents the unperturbed hole energy, i.e. the orbital energy in the ground state. The resonances in the spectrum are found solving the equation E = E ° + Re Znn(E)

(6)

and tile width of the resonance is determined by Im Enn(E) and by the dispersion of Re Znn(E). We close this short display of formulas by mentioning that the core hole spectral function fulfils the sum rule

j-de -2-~A(E)

= 1.

(7)

The distinction between mean field effects and fluctuation effects can be illustrated by the lowest order contribution in perturbation theory to the self-energy o f the hole, which is illustrated schematically in Fig. 1. The diagram (a) is the prototype of diagrams corresponding to a mean field approach. The important feature is that the hole carries the label n straight through the diagram. This implies that all virtual or real excitations always couple to the charge distribution of the hole via the Coulomb interaction. The electron hole excitations pq contribute to the relaxation shift of the level n for virtual pair excitations pq. For a higher excitation energy we may have a finite probability for an energy-conserving

(a)

/b)

Fig. 1. Self-energy contributions corresponding to excitation of an additional electron-hole pair. (a) Relaxation and shake up/off in response to the charge of the hole. (b) Fluctuations o f the hole. transition across the central part of the diagram with two holes n and p and an excited electron q. The important feature is that all processes where the hole line in state n runs through diagrams of all orders are such where the hole acts as an external potential and they could therefore be approximated by some mean field approach such as the method developed by Hedin and Johansson. 5 The type of diagrams of which Fig. l(b) is the simplest example correspond to fluctuations in which the hole makes a transition from n to n' together with the excitation o f an electron-hole pair p'q'. In usual terminology this corresponds to an Auger or CosterKronig type of process. It is clear that diagrams of this type will give additional contributions to the relaxation shift to be added to those obtained from mean field approximations. They will also give rise to additional structure to the shake-up spectrum corresponding to having the hole in state n' and an electron-hole pair in p'q'. In conventional language such terms correspond to a mixture between the configuration with just one hole n in the ground state configuration and such configurations where we have two holes n' and p ' and one excited electron in the state q,.la There are important differences in the symmetry characteristics between processes of type (a) and (b). In diagrams (a) where the hole quantum label n never changes, there is no change in the angular momentum along the hole line and this implies that the intermediate excitation pq has no angular momentum, i.e. it must correspond to a monopole excitation. In processes of type (b) where the hole changes its state from n to n' we may have a change in angular momentum at the vertex. The conservation rules for the Coulomb matrix element demand that the same change appears in the intermediate excitation p'q'. This implies that the hole may now couple strongly to electron-hole pairs of a different symmetry than a monopole. If, e.g. the change in the transition n ~ n' corresponds to Al = 1 we may have

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NEW DYNAMICAL EFFECTS IN CORE HOLES

strong dipolar transitions in the intermediate state. The overall symmetry o f the intermediate state, however, is determined by the hole state n. All processes of type (b) in Fig. 1 correspond to fluctuations in the distribution o f the hole charge. The symmetry is determined by the change in quantum numbers in the virtual transition n ~ n'. If there is no change in angular m o m e n t u m involved we shall have a monopole fluctuation o f the hole. If the transition corresponds to AI = 1 we shall have a dipolar fluctuation around the hole n. This will interact with dipolar fluctuations among the other electrons corresponding in lowest order to virtual excitations o f e l e c t r o n - h o l e pairs p'q', having dipolar symmetry. The two sets o f fluctuations, that o f the hole and that of the other electrons will couple strongly through the Coulomb forces and may give rise to large contributions to the relaxation o f the core level. This interaction will be particularly strong in cases where the dipolar fluctuations of the hole can couple to strong dipolar excitations o f the core. The importance o f fluctuations is to lowest order determined by the energies and interaction matrix elements for the configuration with one hole n in the ground state on one hand and the competing configuration with two holes n' and p ' and one excited electron q'. Different situations may occur. (a) The states n'p'q' correspond to higher excitation energies (a deeper state) than that corresponding to the state with one hole n. The fluctuations o f the hole will give an additional relaxation shift which reduces the binding energy o f the core level. In the satellite region one would see structure due to the various excitations n'p'q'. In this case the relaxed hole corresponds to the most shallow state o f a system with a hole n and is seen as a strong line in the photoelectron spectrum. Even in case o f considerable fluctuations we have a valid quasiparticle picture and a good elementary excitation o f the system. (b) The states n'p'q' o f interest are discrete or partly continuous but fall in the very region of the hole state n. The coupling is now very strong. The relaxed state n may or may not be the lowest state, the relaxation shift may be positive or negative and may even find states predominantly o f the type n'p'q' which occur at lowest excitation energy. This indicates that the system may no longer be stable in the usual sense when we create an initial hole n, but that states predominantly in a different configuration n'p'q' may be more shallow. The validity o f a quasi-particle description is now questionable and the structure o f the relaxed state has to be examined in each individual case. (c) The states n'p'q' o f interest may failin an extended continuum in the region o f the core hole n. This implies that the self-energy has now a finite

167

imaginary part in an energy interval around the unperturbed energy n. This may lead to a complete smearing o f the spectral strength over an appreciable energy interval. If this is the case we see no line at all corresponding to n and there is simply no quasi-particle state o f the N -- 1 electron system which corresponds to the creation o f the initial hole n. The one-electron state n with its orbital properties have then only a formal significance in the theory and do not correspond to a physical excitation. Typical examples o f the situations just described will be given in the following section. 3. APPLICATIONS TO THE 4p SPECTRUM OF Xe AND Ba The 4p and 4s spectra have recently been studied over a range o f elements b y Gelius 9 and Kowalczyk et al. 14 A preliminary theoretical discussion of the 4p spectrum o f Xe was given in references 15 and 16. Here we wish to present further arguments and some results o f a calculation o f the photoelectron spectrum for Xe and, to some extent, Ba. Let us consider the case o f a 4p-hole in Xe or Ba. Figure 1 shows the most important contributions to the self-energy~4p if we choose n = 4p. Figure l(a) describes the normal monopole relaxation process in response to the classical charge o f the 4p-hole leading to a shift o f the core-line from the frozen HF position and to an associated shake u p / o f f spectrum. Solution of equation (6) gives a shift o f the core line o f ~ 6 eV. In this case, relativistic corrections are quite important so that one should add the self-energy correction to the relativistic HF one electron 4pl/2 and 4p3/~ energies. Equivalently, one can perform a relativistic HF ASCF calculation as in reference 9. Comparison with the experimental results of Gelius 9 in Fig. 2 shows that what has been identified as the 4p3/2 line appears 11 eV below the position expected from normal monopole relaxation while the 4pl/2 line seems to be missing altogether. This indicates that there must exist some relaxation mechanism which is a good deal more important than the usual monopole relaxation described by any mean field approximation. Our calculations show that the essential features o f the experimental results are well reproduced by the dipolar fluctuation processes o f the type given in Fig. l ( b ) if we choose n' = 4d. The main contribution comes from the transition (p', q') ~ (4d, rnf) or, in a configuration interaction picture, when we allow the system to make the transitions 4p s ~ 4dSrnf, m discrete or continuous. When m is continuous (the case o f Xe) we are dealing with so called super-Coster-Kronig transitions that can make the 4p-hole decay very rapidly (McGuire17). However, here we want to stress the

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NEW DYNAMICAL EFFECTS IN CORE HOLES

Vol. 19, No. 2

30.000 i

20.000

10.000

4d81 I

4s

210

200

190

100 170 Binding energy

160

150

140

(-E} (eV)

Fig. 2. Photoelectron energy distribution l(e) for Xe in the 4s-4p region. The kinetic energy e = ~o + E, where w is the photon energy. -Experiment. - Present theory. The theoretical 4d 8 double ionization threshold is marked in the figure. The hatched area around 150 eV gives the strength of discrete resonances of the type 4darer when represented on a continuum form.

1.5 (ryz)

//

/ 1

//

,/ /

0~ ~

/'"' I - O . S ~ (e)

/

-15..../ --{~....Z13--/-~2 /~/-~

/'

Id?

../~'
///i

/ (c)

" /'.

/ (b)

/ /-10 E

~1 /(a) (ryd)

/~l

Fig. 3. Real and imaginary parts of the self-energy for a 4p-hole in Xe according to Fig. l(b). Re Z4p(E ). ...... Im Z4p(E). The figure shows the poles in Re Z4~(E) due to 4damf, m = 4 - 7 , excitations and the hatched area symbolizes the remaining levels below the 4d 8 threshold. The straight lines E -- En° denoted by ( a ) - ( f ) give the graphical solutions to equation (6) in a number of cases: (a) Fictitious case: Gives the qualitative behaviour for 4Sl/2 in Xe or 4P3/2 in Ba (quasi-hole strength; width) = (0.80; 1.9 eV); (b) 4pa/2 in Xe; E ° = -- 11.52 ryd; (c) 4pl/2 in Xe; E ° = -- 12.44 ryd; ( d ) - ( f ) Fictitious cases: Gives qualitative behaviour in the case of Coster-Kronig broadening. (quasi-hole strength; width): (d) (0.94; 9.2 eV); (e) (0.88; 5.1 eV); (f) (0.95; 3.6 eV). importance of these processes as virtual transitions, where the corresponding fluctuations of the 4p-hole give rise to an additional shift of the 4p core-level. The f-states in Xe are resonantly located in the 4 p - 4 d region of space. Since all wave functions involved

are localized in the same region, one obtains very large radial overlap and consequently large matrix elements. The same conclusion applies to Ba where the main process involves the 4fstate. Since the main process in both cases is localized inside the core it is rather insensitive to

Vol. 19, No. 2

NEW DYNAMICAL EFFECTS IN CORE HOLES

the particular environment, be it vacuum or a solid. In Fig. 3 we have shown the non-relativistic selfenergy part Z4p(E) for Xe for the fluctuations of the type corresponding to Fig. l(b). The figure also shows the solutions of equation (6) for 4pl/2 and 4p3/2. Fig. l(b) gives only the lowest order diagram and the calculation has been based upon an extended RPA summation to account properly for the high polarizability due to 4d ~ f transitions. Indeed the hole is strongly coupled to a collective motion involving the entire d-shell. The presence of an "extra" 4d hole will modify the 4d ~ f dipole polarizability and this has been taken into account in the calculations. In Fig. 3 we have put the levels denoted by E 4/)1/2 ° and E 4p3/2 ° at the positions corresponding to monopole relaxation so that the structure arises entirely from the dipolar fluctuations of the hole. In the case o f / = 1/2 the solution that would correspond to the relaxed 4pl/2 hole is completely smeared out over an appreciable energy interval and there is no identifiable 4pl/2 hole structure left. This corresponds to the schematic case (c) discussed in section 2. The conclusion is that there is no physical quasiparticle state at all corresponding to a 4pl/2 hole in Xe (and Ba). This is in complete agreement with the experimental finding 9 that the 4pl/2 ESCA peak is missing in Xe and Ba (and several other elements). In the case of the / = 3/2 spectrum the solution for Xe on one hand and Ba on the other appear to be quite different. For Xe there is one solution with great strength corresponding to the main peak at --145.4 eV in Fig. 2. This solution is much closer to the threshold for the 4d8mf excitations than to the unperturbed 4p3:2 hole at E°p~,~. Figure 3 shows that this peak is split off from the 4dSmf excitations and calculations show that it takes up ~ 40% of the total strength of the 4p3/2 spectrum. The interpretation of this analysis is that one should not interpret the main pe,.,zkin Fig. 2 as a 4p3/2 peak but as a strongly coupled mode of the 4 d 8 4 f excitations and a 4p3/2 hole. Therefore like in the 4p~/2 case we find that there is no quasi-particle state corresponding to a 4p3/2 state. The strong elementary excitation seen is a resonant mode coupling the dipolar fluctuation of the 4p hole to the strong dipolar collective mode arising from the transitions 4d ~ mr. The situation for Ba is simpler. The dominant dipolar mode comes from the transition 4d -~ 4 f and gives rise to an additional relaxation shift of ~ 13 eV. However the 4 d 8 4 f state now lies deeper than the unperturbed 4P3/2 and thus the relaxed state will have an appreciable component of 4p. Although the effects of dipolar fluctuations is large for Ba as indicated by the strong relaxation shift, we feel that the quasi-particle concept of a 4p3/2 hole in Ba is valid.

169

These examples illustrate the effects that occur in systems where the interaction effects involving the hole are strong. In such cases certain expected quasi-particle like hole states may not exist. This was illustrated by the 4pl/2 and 4p3/2 spectra of Xe. The 4pl/2 immediately decays into a continuum of states. The expected 4p3/2 state is replaced by a strong excitation where the fluctuating 4p hole is strongly coupled to a strong dipolar collective excitation of the d shell forming a resonant mode of rather narrow width. The possibility of such strong couplings between a fluctuating hole and a collective excitation is expected to occur also in other atomic and molecular systems. The conditions are however quite selective as is seen from the fact that the corresponding 4p3/2 excitation in Ba, just two atomic numbers away, shows a regular quasi-particle behavior. Finally we include the dipole matrix elements of equation (4) in order to get the rate of photoelectrons with energy e hitting the detector. The proportionality constant depends on the absolute flux of photons, the number of atoms in the sample etc. We adjust the proportionality constant in the 4s-region by demanding that the theoretical 4s-peak have the same area as the experimental 4s-peak minus a background which certainly comes from the 4p-region but the origin of which is not clear at present, except that it is unlikely to be 4p shake off. The theoretical result is shown in Fig. 2 and the reasonable agreement with experiment in the entire 4s and 4p region suggests that the basic relaxation mechanisms have been well accounted for. A detailed description of our calculations with further applications will be presented elsewhere. In this paper we have only discussed the properties of 4s- and 4p-core holes. However, similar effects are certainly present for 3s, 3p and 5s-holes in e.g. Xe. Further examples are 3s-holes in the transition metals. Bagus and coworkers la have recently performed CI calculations for Mn 2÷, fnding that a 3s-hole is strongly perturbed by configuration interaction involving transitions of the type discussed in this paper [Fig. l(b)]. 4. CONCLUSION The applications to Xe and Ba seem to give strong support that dynamical effects play an important role in the relaxation of core holes in heavier elements. The advantage of the method is that one calculated directly the photoelectron flux and avoids explicit calculations using configuration interactions. It seems that the data recently given for the elements from Z = 42 to Z = 75 by Kowalczyk et a/. 14 and by Gelius 9 are explained in the same way and detailed calculations of the actual spectral shape are in progress. It is also clear that the effects are by no means limited to the 4p level structure but depend only on the favorable circumstances of

170

NEW DYNAMICAL EFFECTS IN CORE HOLES

strong overlap giving rise to large matrix elements and transition probabilities. Strong effects of fluctuations should therefore occur quite frequently for heavier

Vol. l 9, No. 2

elements, free or bound in solids and they are expected to occur in certain molecules as well.

REFERENCES 1.

KOOPMANS T., Physica 1,104 (1934).

2.

SUREAU A. & BERTHIER G., J. Phys. (Paris) 24, 672 (1963).

3.

BAGUS P.,Phys. Rev. 139, A619 (1965).

4.

LINDGREN I.,Ark. Fys. 31, 59 (1966).

5.

HEDIN L. & JOHANSSON A., J. Phys. B2, 1336 (1969).

6.

SLATER J.C. & WOOD J.H., Int. J. Quant. Chem. Symp. 4, 3 (1971).

7.

GOSCINSKI O., PICKUP B.T. & PURVIS G., Chem. Phys. Lett. 22, 167 (1973).

8.

BASCH H., J. Electr. Spectr. 5,463 (1974).

9.

GELIUS U., J. Electr. Spectr. 5,985 (1974).

10.

SIEGBAHN K., J. Electr. Spectr. 5, 3 (1974).

11.

SHIRLEY D.A.,Adv. Chem. Phys. 23, 85 (1973).

12.

HEDIN L. & LUNDQVIST S., Solid State Phys. 23, 1 (1969).

13.

BAGUS P.S., FREEMAN A.J. & SASAKI F., Phys. Rev. Lett. 30, 850 (1973).

14.

KOWALCZYK S.P., LEY L., MARTIN R.L., MCFEELY F.R. & SHIRLEY D.A., Faraday Discussions of the Chemical Society. Vancouver, B.C., July 15-17 (1975).

15.

LUNDQVIST S. & WENDING., J. Electr. Spectr. 5, 513 (1974).

16.

WENDIN G. & OHNO M., Proc. Int. Conf. on Electron Spectroscopy, Kiev, 3 - 8 June (1975).

17.

MCGUIRE E.J.,Phys. Rev. A9, 1840 (1974).