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Particle–hole interaction effect on an Auger-electron spectrum
Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69–79 www.elsevier.nl / locate / elspec
Particle–hole interaction effect on an Auger-electron spectrum Masahide Ohno Quantum Science Research 2 -8 -5 Tokiwadai, Itabashi-ku, Tokyo, 174 -0071 Japan Received 13 November 2000; accepted 9 February 2001
Abstract Auger-electron spectroscopy (AES) spectral-lineshape governed by a final-state interaction of a localized two-hole (2h) state with a three-hole one-particle (3h1p) continuum or bound state, is studied by a Green’s function method. The particle–hole interaction U in a 3h1p state can affect the AES spectral-lineshape significantly. Even when the coupling between a 2h state and a 3h1p continuum is strong, there can be a sharp resonant 2h state. When the coupling is ‘‘weak’’, the quasi-particle approximation for a 2h state can break down. As an example, the M 45 –N 23 N 45 AES spectra of elements Pd to Ba are studied. The effect of particle–hole interaction on an AES spectral-lineshape of a charge transfer system is also discussed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Particle-hole interaction; Auger-electron spectroscopy; Charge transfer system
1. Introduction Recently a many-body theory has been employed to study the effect of the particle–hole interaction U in a two-hole one-particle (2h1p) state on a corelevel XPS spectrum [1–3]. A 2h1p state is coupled with an initial 1h state by Auger or Coster–Kronig (CK) transition. With an increase of U, the density of particle states in a 2h1p state surges towards the double-ionization threshold. The density of states becomes narrow and dense. When U becomes large, a Coulomb-bound particle state splits off from a continuum and appears below the double-ionization threshold. The interaction between an initial 1h state and a 2h1p state essentially depends on their separation-energy and coupling strength. Thus if an initial 1h state approaches the double-ionization threshold with an increase of U, the interaction strength increases dramatically. When U becomes very large, the interaction becomes essentially that
between an initial 1h state and an unperturbed Coulomb-bound 2h1p state. Thus the initial core– hole spectral lineshape can depend critically on U. As an example, a many-body model calculation is performed to provide an overall description of the 4s- and 4p-hole XPS spectra of Pd to La [2,3]. We shed light on the mechanism of the anomalous spectral-lineshape change with the atomic number. In an interacting many-electron system, doubly subtracted electrons or two quasi-particles could be simply not a stable component of the system: They could and would decay into two or a number of constituent parts and the system’s true exact excitation spectrum is not a 2h-like at all. Moreover, the state is no longer a 2h bound-state because of the finite lifetime-effect. In general such a 2h state can be interpreted in terms of a resonance and 2h boundstates embedded in a 3h1p continuum. In fact it is entirely possible for a real, stable 2h bound-state to exist even when the coupling is strong. What inter-
0368-2048 / 01 / $ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 01 )00274-2
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
ests us most about a 2h bound state embedded in a 3h1p continuum is, not so much the possibility of its existence, as the fact that if it can be formed in the presence of strong coupling, then a fortiori it must be possible to produce a sharp resonance. In other words, the presence of a sharp resonance is not necessarily restricted to a regime of weak coupling. To demonstrate the many-body effect in a 2h state, we employ a many-body model calculation approach. We study the effect of the particle–hole interaction U in a 3h1p state on a 2h state. The density of 2h states depends critically on U. Thus the AES spectral-lineshape can be governed by U. As examples we consider the M 45 –N 23 N 45 AES spectra of Pd to Ba, which show anomalous lineshapechanges with the atomic number [4–7]. We provide an overall description of the density of final states changes with the atomic number. We consider also a similar particle–hole interaction effect on an AES spectrum of a charge transfer (CT) system.
2. Density of two-hole states
2.1. Theory We consider a 2h state (ij) in which one of the holes, i, involves a strong electron-correlation so that a 2h state interacts with 3h1p continuum states ( j 22 1 21 ´). Here we consider the ‘‘hole-hopping’’ relaxa–1 21 21 tion (screening), namely i –j 1 ´ in the presence of an extra j hole. The density of perturbed 2h states is 1 D rf (v ) 5 ] ? ]]]]]]]]] p (v 2 ´i 2 ´j 1 Uij 2 L)2 1 D 2
(1)
Here ´i (´j ) is the energy of hole i ( j). The bare hole–hole interaction, Uij, is included to infinite order by the ladder approximation. L and D provide the energy-shift and lifetime-broadening of a 2h state by the coupling with 3h1p states, respectively. L and D are F 2 U(F 2 1 D 2 ) L 5 2uV u 2 ]]]]]] (1 2 UF )2 1 (UD)2
(2)
D D 5 2uV u 2 ]]]]]] (1 2 UF )2 1 (UD)2
(3)
Here F(v ) and D(v ) are the real part and imaginary part of the unperturbed 3h1p propagator Go , respectively V is the energy-independent coupling matrix element between a 2h state and 3h1p states. The factor of two comes from the relaxation of hole i and the screening of Uij by the aforementioned ‘‘hole-hopping’’ relaxation. The unperturbed 3h1p propagator Go is
E
r (´) Go (v ) 5 ]]]]] d´ v 1 ´ 2 ´j 2l 2 id
(4)
´, v and ´j 2l are the particle-state energy, the 2h energy and the 3h energy, respectively. r is the density of unperturbed particle-states. Here unperturbed means without the screened particle–hole interaction U in the 3h1p states. The holes in the 2h and 3h1p states are assumed to be atomic-like localized so that the hole–hole interaction energy in the 3h1p states is taken into account by introducing the effective 3h energy. We obtain L and D by renormalization of 2 uV u 2 Go by including the screened particle–hole interaction U in the 3h1p state, to infinite order by the ladder and ring approximation. Here screened means the screening of the bare particle–hole interaction in terms of the particle–hole pair excitations. U is assumed to be particle-energy independent. We approximate the density of unperturbed particle-states by a parabolic band.
F S
3 ´ 2W r (´) 5 ] 1 2 ]] 4W W otherwise r (´) 5 0
D G for 0 # ´ # 2W, 2
(5)
r is normalized is 1.0. 2W is the width of density of the unperturbed particle-states. Introducing E 5 W 1 v 2 ´j 2 l we rewrite Eq. (1) 1 D rf (E) 5 ] ? ]]]]]] p (E 2 Eo 2 L)2 1 D 2
(6)
Here Eo 5 ´i 1 ´j 2 Uij 2 (´j 2 l 2 W ), is the unperturbed 2h state energy relative to the center of the 3h1p states (E50). Here unperturbed means without the coupling with the 3h1p states of interest. E 5 W corresponds to the 3h state energy v 5 ´j 2 l In the M 45 –N 23 N 45 Auger transition of Pd to Ba, the 4p4d (2h) state is coupled with the 4d 23 ´f (3h1p) states. The relativistic average-energies for
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
´i 1 ´j 2 Uij (4p 21 4d 21 2h state) and ´j 2l (4d 23 3h state) of metallic elements Ag to Ba are calculated from the relativistic atomic D SCF hole energies by using the atom–metal energy-shifts estimated by the method described in Ref. [8]. Thus the energy-shift of Eo by the ‘‘no-hole hopping’’ relaxation (screening) is already taken into account by the D SCF approximation. We consider explicitly the hole-hopping relaxation (screening) by taking into account the coupling between the 2h (4p4d) state and the 3h1p (4d 23 ´f) states. The AES spectrum is essentially a superposition of the density of the 2h states, separated by the initial core-level spin–orbit splitting, multiplied by the Auger-transition matrix element and statistically weighted by the ratio of the initial core–electron population. We emphasize that we are not aiming to describe the details of the experimental AES spectral lineshapes. We are more interested in the origin of
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the anomalous AES spectral-lineshape changes with the atomic number. Thus for the sake of clarity, we study the density of the 2h states rather than the AES spectral lineshape. Thus in Figs. 1 and 2 we present L, D, E 2 Eo and the density of 2h states (Eq. (6)), calculated for different U, Eo and V values (W53.0 is fixed). E is in units of Rydberg.
2.2. Results and discussion 2.2.1. M45 –N23 N45 Auger transition of Pd to Ba In Fig. 1 we show the density of 2h states with a change of U and Eo when V50.707. Fig. 1a (U 50 and Eo50) and Fig. 1b (U 50.3 and Eo50.5) correspond to Ag and In, respectively. U is small and Eo lies near the center of the 3h1p continuum states. There is a Es which satisfies Es 2 Eo 2 L (Es)50. However, dL(E) / dE at Es is smaller than 1.0 and is positive. The renormalization factor for the
Fig. 1. The density of two-hole states, rf , obtained by the following parameter values (V50.707 and W53.0 are fixed). (a) Ag: U 50.0. Eo50.0, (b) In: U 50.3, Eo50.5, (c) Sb: U 50.5, Eo51.3, (d) Te: U 50.7, Eo51.8, (e) I: U 51.4, Eo52.2, (f) Xe: U 52.1, Eo54.3, (g) Ba: U 52.5, Eo54.6. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Fig. 2. The density of two-hole states, rf , obtained by the following parameter values (V50.354 and W53.0 are fixed). (a) U 50.0, Eo50.0, (b) U 50.3, Eo50.5, (c) U 50.5, Eo51.3, (d) U 50.7, Eo51.8, (e) U 51.4, Eo52.2, (f) U 52.1, Eo54.3, (g) U 52.5, Eo54.6. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
density of 2h states, Z 5 (1 2 dL(E) / dE)21 , is larger than 1.0 at Es. Z provides the probability that the initial 2h state remains in the same state because 2 dL(E) / dE provides, to the lowest order in the perturbation theory, the probability that the initial 2h state makes a transition to a 3h1p state. When Z is negative or larger than 1.0, the ‘‘solution’’, Es,does not have any physical meaning and Z cannot be defined. To describe the density of 2h states by the Breit– Wigner resonance-lineshape, namely a resonant 2h state in 3h1p continuum states, a number of specific assumptions must be fulfilled. The most important of them is that the width and the shift from Es to Er (Er is the resonance energy) are both small relative to the energy scale on which the other quantities in the excitation amplitude vary appreciably. If the shift, Er 2 Es, is not small, the Taylor expansion around Es to obtain Z is no longer useful near Er. The shift
is given by 2 D(E)xZ 2 (dD(E) / dE) at Es. Thus if D(E) varies slowly (or is constant) and is not large, then Er 2 Es is small (or zero) and the Taylor expansion is valid. Otherwise, Er deviates substantially from Es. Z cannot be defined at all. If the width (D(E)) is not small, the resonance peak is superposed upon a background that varies as rapidly as the peak itself. Its shape may then be quite distorted; in fact, it may effectively become invisible. In the present case D(E) is rather constant around Es. Thus the shift, Er 2 Es, is small. However, D is large. Thus the conditions for the Breit–Wigner resonance-lineshape are not really satisfied. Thus it is appropriate to say that the notion of quasi-particle becomes illusory. Es is close to Eo. Thus the Auger-electron energy evaluated by the D SCF approximation (including the atom-metal energy shifts) becomes close to the center of the AES spectral lineshape. The relaxation
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
energy-shift of the 2h state, which is approximately Es 2 Eo, is small but the damping is large. The time scale of the hole decay is much shorter than that of the relaxation so that the relaxation becomes incomplete. The screening of the hole–hole interaction becomes incomplete because of absence of the holehopping screening. It should be noted that despite that one of the holes (4p hole) decays very rapidly, the hole–hole interaction in the 4p4d state does not become zero. Figs. 1c and 1d correspond to Sb and Te, respectively. With an increase of U, the density of the particle states in the 3h1p states begins to surge towards the triple-ionization threshold. The unperturbed 2h state begins to approach to the maximum of the density of the 3h1p states. The interaction between the unperturbed 2h state and the 3h1p states becomes large. D(E) is large and varies appreciably around Es. In Fig. 1c Er 2 Es becomes large. Thus the conditions for the Breit–Wigner resonancelineshape are not really satisfied for both cases. Thus it is appropriate to say that the notion of quasiparticle becomes illusory. Around Es, E 2 Eo and L becomes almost parallel. Then the density of 2h states becomes approximately 1 / hpDj. This explains a dip in the density in Figs. 1c and 1d. One may see the interaction between a 2h state with a 3h1p states as a kind of dissociative broadening. The dissociative broadening is the broadening of a localized 2h state in an alloy. The broadening arises from the mixing with delocalized band-states. In alloys a 2h bound state may mix with states with 1h in a host band and 1h in a partner-element band. The broadening becomes substantial when the host two-electron binding energy is in a large density of states region of ‘‘mixed convolution’’ of the host band with the partner band [9,10]. Fig. 1e corresponds to I. With an increase of U, the density of the particle states surges dramatically near the triple-ionization threshold. At the same time the unperturbed 2h state approaches the threshold. Thus the strength of the interaction between the unperturbed 2h state and the 3h1p states increases significantly. As a result of the strong interaction between the unperturbed 2h state and the 3h1p continuum states, an unperturbed 2h bound state splits off and appears below the threshold. The rest of original strength of the 2h state goes to the 3h1p
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continuum states and the quasi-particle approximation breaks down. In a strong coupling regime a 2h bound state splits off as a sharp resonant state for which the quasiparticle approximation is valid. Es coincides with Er because D(E) 5 0. The relaxation energy shift Es(Er)2Eo is substantially large. The relaxation is complete because the decay time is much longer than the relaxation time. The relaxation energy-shift includes the relaxation energy-shift of a hole and the energy-shift of the hole–hole interaction energy by the ‘‘hole-hopping’’ screening. Figs. 1f and 1g correspond to Xe and Ba, respectively. With an increase of U, an unperturbed Coulomb-bound particle state splits off from the continuum and appears below the ionization threshold. Such a bound state dominates in the density of the particle states. As a result, the density of 2h states is essentially governed by the interaction between the 2h state and the 3h1p bound state. The unperturbed 2h state lies below the 3h1p bound state. Thus the perturbed 2h bound state (Es) which fulfills Es 2 Eo 2 L (Es) 5 0, is a Koopmans-theorem (KT)-like 2h state. The rest of the original strength of the 2h state goes to the 3h1p continuum. The quasi-particle approximation breaks down for a 2h state in the 3h1p continuum. If an unperturbed 2h state lies above an unperturbed Coulomb bound 3h1p state, then a bound state (Es) which fulfills Es 2 Eo 2 L (Es) 5 0 and D(Es)50. becomes a perturbed 3h1p state. The existence of an unperturbed 3h1p bound state is governed by the potential-binding condition 1 2 UF 5 0 and D 5 0, while that of a perturbed 3h1p bound state is governed by E 2 Eo 2 L(E) 5 0 and D(E)50. The former state is a split-off state which depends on the ratio of U to the continuum bandwidth, while the latter is a state which depends on the relative energy Eo. The mechanism for the appearance of the former state differs from that of the latter state. When the density of 3h1p states becomes a narrow and dense band with an increase of U, and it becomes degenerate and interacts strongly with an unperturbed 2h state, a perturbed 2h bound-state splits off (Fig. 1e). It is incorrect to interpret the split-off mechanism as that of the split-off of a localized 2h state when 2h are created in a narrow
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
band. A localized 2h state arises when the potentialbinding condition 1 1 UF 5 0 and D50 are fulfilled. Here F and D are the real part and imaginary part of the two-hole propagator without the hole–hole interaction U, respectively. U is the hole–hole interaction energy. The split-off of a perturbed 2h bound state from the band depends on the relative energy Eo and the coupling strength V, namely on the interaction between two different states, not on the potentialbinding condition, while the split-off of an unperturbed 3h1p bound state from the band (Figs. 1f and 1g) depends on the potential-binding condition and is quite analogous to that of the appearance of a localized 2h state in the C2VV AES spectrum of a narrow-band metal. The potential-binding condition is governed by the particle-hole interaction in a 3h1p state. The present many-body calculation reveals the mechanism of anomalous spectral-lineshape changes of the M 45 –N 23 N 45 AES spectra of Pd to Ba with the atomic number. With U increase, the density of particle states in the 3h1p states surges towards the triple-ionization threshold. The density of particle states becomes narrow and dense. When U becomes
large, a Coulomb-bound particle state splits off from the continuum and appears below the threshold. Such a bound state dominates in the density of particle states. U increases with the atomic number. At the same time an unperturbed 2h state approaches to the triple-ionization threshold (see Fig. 3). As a result the interaction between an unperturbed 2h state with a 3h1p continuum and / or bound state increases. It is U which increases predominantly the coupling strength between a 2h state and 3h1p states with the atomic number. This most important point was not realized in a theoretical study of the M 45 –N 23 N 45 AES spectra of Pd to Xe by the ab-initio atomic Green’s function calculation by the author [4–6]. The present model calculation has an advantage that one can monitor the variations of the strength of physical quantities which govern predominantly the spectral-lineshape change. The major problem neglected so far is the multiplet structure of the 2h (4p4d) state. The present calculation shows that except for the case of a 2h bound state, the quasi-particle approximation for a 2h state breaks down. In such a case the concept of the multiplet splitting becomes illusory because the
Fig. 3. Theoretical relativistic atomic D SCF energies of single- and multiple-hole states of the elements Z546 to Z557.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
decay width is much larger than the multiplet splitting. This is related to the fact that delocalized holes do not form multiplet-coupled states. One may explain in an qualitative manner the density of 2h states as the 4p hole spectral-function of the Z 1l atomic element. The 4p hole in the presence of a 4d hole may be interpreted as that of the Z 1 1 atomic element. The density of the 3h1p states becomes rather similar to that of the 2h1p states of the Z 1 1 atomic element because of an increased U by the presence of an extra hole. The parameter value U for the 3h1p state is indeed larger than that for the 2h1p state. Moreover, it is closer to U for the 2h1p state of the Z 1 1 atomic element. However, the separation energy between the 2h state and the 3h1p states differs from that between the 1h state and 2h1p states of the Z 1 1 atomic element because of an increased hole–hole interaction in the 3h1p state by the presence of an extra 4d hole. Thus strictly speaking the Z 1 1 approximation is not valid. One cannot simply apply the Z 1 1 approximation for a transition which involves multipleholes. So far we considered the case when one hole in a 2h state (i ± j) induces a strong electron-correlation. We can treat also the case when both holes in a 2h state involve strong electron-correlations. When i 5 j, in general the factor of two in Eqs. (1)–(4) becomes four. However, when i 5 j 5 4p, the hole-hopping CK process of each 4p hole occurs, but the dipolar screening of the hole–hole interaction by the CK process does not occur. Thus the factor of two remains in Eqs. (1)–(4), One cannot simply approximate the density of 4p 22 (2h) states by a self-convolution of the 4p spectral function of the Z 1 1 atomic element because of the presence of the hole–hole interaction and many-body effects described above. So far there is no experimental study of the M 45 –N x N y (X5Y5 4s, 4p) AES spectra of Pd to Ba. In the M 45 –N 1 N 23 Auger transition, the 4s 21 4p 21 (2h) state interacts with the 4p 22 4d 21 ´f (3h1p) states. The 4p 22 4d 21 (3h) state interacts with the 4p 21 4d 23 ´f (4h1p) states. The cascade interaction continues. Thus the many-body effect becomes much more complicated than that of the M 45 –N 23 N 45 Auger transition. It would be interesting to investigate the M 45 –N x N y AES spectra of Pd to Ba by Auger-
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photoelectron coincidence spectroscopy (APECS) because the density of final 2h states can be directly obtained when the Auger electron is measured in coincidence with a selected initial core–hole state. The 4s XPS spectrum looks quite normal, except for a large relaxation energy shift from the D SCF hole energy, caused by 4s 21 –4p 21 4d 21 ´f CK virtual process [2]. When U is very small, the imaginary part of the 4s-hole self-energy is given by the convolution of the density of unperturbed particle states (´f ) with the density of the perturbed 2h (4p4d) states, multipled by the energy-independent CK transition matrix element. With an increase of U, the convoluted density will be perturbed by U. In general, the lifetime broadening of 2h states is negligible, compared to the density of particle states. Then the density of 2h1p states becomes that of particle states perturbed in the presence of two holes. This is the case with the 4p-hole self-energy. However, this is not the case with the 4s-hole self-energy. The density of perturbed 2h (4p4d) states is comparable with the density of particle states (´f ). Then the density of unperturbed 2h(4p4d)1p(´f ) states, given by a convolution of the density of unperturbed particle states (´f ) with the density of perturbed 2h (4p4d) states, is much broad, compared to the density of unperturbed particle states. Both the relaxation energy-shift and lifetime broadening of a 4s hole can be affected by the broadening of the density of 2h(4p4d)1p(´f ) states. The 4s hole involves a much more complicated many-body effect than a 4p hole, despite that the former hole spectrum looks much more normal than the latter one. The many-body effect does not manifest in the ionization spectral-lineshape.
2.2.2. V50.354 case Eo lies in the 3h1p continuum (Figs. 2a–2d). However, because of a small coupling strength, D becomes small and rather constant around Es so that the energy shift, Er–Es, becomes small. As a result the density of 2h states becomes a resonant-like lineshape. It is appropriate to say that the quasiparticle approximation for the 2h state is valid. When Eo lies below a Coulomb-bound 3h1p state (Figs. 2f and 2g), the relaxation energy-shift of the perturbed 2h bound state becomes smaller and its spectral
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
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intensity becomes larger because of a smaller coupling strength. The one-electron picture becomes valid.
3.1. Theory For the sake of simplicity we consider the case when two localized-holes are created in the same inner-valence level in an acceptor. The density of the perturbed 2h states is (7)
Here v0 is the unperturbed 2h state energy without the CT interaction. They are F 1 U(F 2 1 D 2 ) L 5 4uV u 2 ]]]]]] (1 1 UF )2 1 (UD)2
(8)
D D 5 4uV u 2 ]]]]]] (1 1 UF )2 1 (UD)2
(9)
V is the energy-independent coupling matrix element. F and D are the real part and the imaginary part of the unperturbed 3h1p propagator, respectively. The unperturbed 3h1p propagator is
r (´) E]]]]]] d´ 5 2 F 1 iD ´ 1 v 2 ´ 2 v 2 id a
d
D G for 0 # ´ # 2W, 2
(11)
Then we introduce E 5 vo 2 v 2 W 1 ´d and Eo 5 ´d 2 W. From Eq. (7) we obtain the density of perturbed states two-hole.
3. Charge transfer system
1 D r (v ) 5 ] ? ]]]]]] p (v 2 v o 2 L)2 1 D 2
F S
3 ´ 2W ra (´) 5 ] 1 2 ]] 4W W otherwise ra (´) 5 0
(10)
o
Here unperturbed means without the particle–hole interaction in the 3h1p CT state. We renormalize the unperturbed 3h1p propagator (Eq. (10)) by including the screened particle–hole interaction of U, to infinite order by the ladder and ring (low density) approximation. Then multiplying by uV u 2 , we obtain the real part (Eq. (8)) and the imaginary part (Eq. (9)) of the renormalized term which provides the CT interaction between the 2h state and the 3h1p CT states. The factor of four in Eqs.8 and 9 comes from the screening of each inner-valence hole and that of the bare hole–hole interaction between two innervalence holes. We approximate ra by a parabolic band given by
1 D r (E) 5 ] ? ]]]]]] p (E 2 Eo 2 L)2 1 D 2
(12)
E 5 Eo corresponds to v 5 vo , while E 5 W(2W ) does to v 5 ´d 1 vo 2 2W (v 5 ´d 1 vo ), namely the unperturbed 3h1p state in which there is CT from the donor to the top (bottom) of the unoccupied acceptor-band in the presence of two valence-holes. For the density of perturbed 2h states, the energy scale E is the negative of 2h energy. r obtained for different U and Eo are shown in Figs. 4 and 5, together with L, D and E 2 Eo.
3.2. Results and discussion Figs. 4a–4d correspond to the cases when there is an excitation predominantly from below the Fermi level. When U is small (Figs. 4a and 4b), the smallest binding energy state is a Koopmans theorem (KT)-like two-hole state, in which a screening electron still remains predominantly in the occupied part, i.e. there is almost no extra screening. The density of KT-like 2h state increases with a decrease of V because of a smaller coupling strength. The continuum state is a shakeup state, in which an electron is excited from the occupied part to the unoccupied part of the resonance density of states. The quasi-particle approximation breaks down for the satellite. With an increase of U (Figs. 4c and 4c), the density of the unoccupied states surges toward the Fermi level. When U becomes larger than the width of the unoccupied part of the resonance density of states, an empty bound-level splits off from the band states and appears below the Fermi level. When an empty bound-level in the acceptor still lies just below the Fermi level (Fig. 4c), the smallest binding energy state is a KT-like state, in which a screening electron still remains predominantly in the occupied part, i.e., no extra hole screening. The density of a
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Fig. 4. The density of the two-hole states, r, obtained by the following parameter values. W 5 1.0, V 5 0.5, Eo5 22.0, (a) U 5 0.0, (b) U 50.3, (c) U 50.8. (d) U 52.0. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
KT-like state increases with a decrease of V because of a smaller coupling strength. In Fig. 4c the CT interaction between the occupied part and the empty bound-level in the acceptor is still not large enough so that the satellite is still a shakeup state, in which an electron is excited from the occupied part to the unoccupied part of the resonance density of states. The quasi-particle approximation becomes valid for the satellite. When U becomes larger (Fig. 4d), an empty bound-level in the acceptor appears predominantly below the occupied part. Then the smallest binding energy state is a CT shakedown state, in which the empty bound-level in the acceptor is predominantly occupied by an electron from the occupied part. There is an extra screening. There is a KT-like state in which a screening electron still remains pre-
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Fig. 5. Same as Fig. 4, except for Eo5 21.0.
dominantly in the occupied part. It is a bound state, the lineshape width of which is due to the valencehole lifetime. With a decrease of U and V, the density of the KT-like state increases. There are shakeup states of non-negligible intensity. Figs. 5a–5d correspond to the cases when there is an excitation predominantly from the Fermi level. When U is small (Figs. 5a and 5b), the behavior of DOS is still similar to that in Fig. 4a and 4b. When U is large (Figs. 5c and 5d), the KT-like state becomes a resonant state. The lineshape-width is nothing but the effective excitation rate at the resonance energy (except for the valence-hole lifetime-broadening). Thus the effective CT coupling strength at the resonance can be determined from the lineshape of the density of 2h states. There is also a shakedown state, in which an electron is transfered from the occupied part to an empty bound-level for the hole screening. With a decrease of V, the density of the KT-like state increases. With an increase of U, the
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
density of the KT-like state decreases. However, when U becomes comparable with the bandwidth of the unoccupied part of the resonance density of states, an empty bound-level splits off from the band. With an increase of U, the density of the KT-like resonant state increases. The major difference between the AES spectra discussed in Section 2.2 and the present one is, that in the latter the unperturbed 2h state always lies below or at the edge of the 3h1p states. The case discussed for Fig. 1f is very similar to that of Fig. 4c. The case when a 3h1p bound state lies below an unperturbed 2h state discussed in Section 2.2 is found in Fig. 4d in which the perturbed 3h1p bound state lies below the perturbed 2h state. The case shown in Fig. 2e is close to that in Fig. 5c, In the former case, because of small ratio of (U /W )50.47 compared to that (0.8) in the latter case, there is no unperturbed 3h1p bound state below the band so that a 3h1p bound state does not split off, while in Fig. 5c there is an unperturbed 3h1p bound state so that a 3h1p bound state splits off. Both Figs. show the presence of a resonant KT-like 2h state. The width of the resonant-like state spectral-lineshape is governed by the coupling strength between the 2h state and the 3h1p state. Thus it is self-explanatory that the same mechanism is governing both spectra.
4. Conclusion By using a many-body model calculation, we study the effect of the particle-hole interaction, U, in 3h1p states on a 2h state. With an increase of U, the density of particle states in the 3h1p states will be pushed towards the triple ionization threshold. The density of particle states becomes narrow and dense. If an unperturbed 2h state shifts towards the threshold at the same time, the interaction between an unperturbed 2h state and perturbed 3h1p states increases dramatically so that a 2h bound state splits off from the band and appears below the threshold. A sharp resonant 2h state appears even in a strong coupling regime, while the quasi-particle approximation may break down in a ‘‘weak’’ coupling regime. When U becomes large, a Coulomb-bound particle state splits off and appears below the threshold. It
takes a large part of the density of particle states. The interaction between a 2h state and 3h1p states becomes essentially that between a 2h state and a 3h1p bound state. As a result, the density of 2h states dominantly consists of discrete states. The present model calculation can provide an overall description of variations of the density of 2h states of the M 45 –N 23 N 45 Auger transition of the elements Pd to Ba. U increases with the atomic number and causes an anomalous AES spectral behavior. The many-body effects associated with the ionization of 4s- and (or) 4p- electrons are still not fully investigated, despite that they provide very unique spectral behaviors. In particular, the cascade decay of 4p(4s) hole in a multiple-hole state may provide unique opportunities to study the post-collision interaction effect on an Auger-electron spectrum, because of a sudden change in the Coulomb potential a slow photoelectron (an Auger-electron) experiences. Further experimental studies using Auger-electron spectroscopy and / or Auger-photoelectron coincidence spectroscopy are needed. The effect of particle–hole interaction on an AES spectrum of a CT system is discussed. When the interaction of an electron transfered from the occupied part to the unoccupied part of the resonance density of states perturbed by two inner-valence holes created by core–hole decay, becomes larger than the bandwidth of the unoccupied part of the resonance density states, an empty acceptor boundlevel splits off from the band. This is governed by the potential-binding condition. When such an empty acceptor bound-level is pulled below the occupied part, a shakedown state appears as a screened two-hole state. When there is an excitation predominantly from the occupied part close to or at the Fermi level, there is a possibility that a KT-like 2h state appears as a resonant state, the width of which is determined by the sum of the excitation rate at the resonance and the valence-hole lifetime-broadening. In such a case one can monitor the excitation rate change with a chemical environmental change. The mechanism which governs the AES spectral-lineshape of a CT system considered here is the same as that proposed for the M 45 –N 23 N 45 AES spectral-lineshape changes with the atomic number. So far there appears to be no relevant experimental observation for a CT system. However,
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Auger-photoelectron coincidence spectroscopy (APECS) will be useful because it reveals directly the density of final 2h states.
References [1] M. Ohno, J. Elect. Spectr. Rel. Phenom. 106 (2000) 37. [2] M. Ohno, J. Elect. Spectr. Rel. Phenom. 107 (2000) 113. [3] M. Ohno, J. Elect. Spectr. Rel. Phenom. 119 (2001) 19–27.
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[4] M. Ohno, G. Wendin, Solid State Commun. 39 (1981) 875, Energies plotted in the spectra of this reference and Ref. [5] The SCF are not accurate. [5] M. Ohno, J.-M. Mariot, J. Phys. C14 (1981) L1133. [6] M. Ohno, Phys. Rev. A38 (1988) 3473. [7] S. Aksela, H. Aksela, T.D. Thomas, Phys. Rev. A19 (1979) 721. [8] M. Ohno, G. Wendin, J. Phys. C15 (1982) 1787. [9] P.A. Bennett, J.C. Fuggle, F.U. Hillebrecht, A. Lenselink, G.A. Sawatzky, Phys. Rev. B27 (1983) 2194. [10] M. Ohno, J. Elect. Spectr. Rel. Phenom. 107 (2000) 103.
Particle–hole interaction effect on an Auger-electron spectrum Masahide Ohno Quantum Science Research 2 -8 -5 Tokiwadai, Itabashi-ku, Tokyo, 174 -0071 Japan Received 13 November 2000; accepted 9 February 2001
Abstract Auger-electron spectroscopy (AES) spectral-lineshape governed by a final-state interaction of a localized two-hole (2h) state with a three-hole one-particle (3h1p) continuum or bound state, is studied by a Green’s function method. The particle–hole interaction U in a 3h1p state can affect the AES spectral-lineshape significantly. Even when the coupling between a 2h state and a 3h1p continuum is strong, there can be a sharp resonant 2h state. When the coupling is ‘‘weak’’, the quasi-particle approximation for a 2h state can break down. As an example, the M 45 –N 23 N 45 AES spectra of elements Pd to Ba are studied. The effect of particle–hole interaction on an AES spectral-lineshape of a charge transfer system is also discussed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Particle-hole interaction; Auger-electron spectroscopy; Charge transfer system
1. Introduction Recently a many-body theory has been employed to study the effect of the particle–hole interaction U in a two-hole one-particle (2h1p) state on a corelevel XPS spectrum [1–3]. A 2h1p state is coupled with an initial 1h state by Auger or Coster–Kronig (CK) transition. With an increase of U, the density of particle states in a 2h1p state surges towards the double-ionization threshold. The density of states becomes narrow and dense. When U becomes large, a Coulomb-bound particle state splits off from a continuum and appears below the double-ionization threshold. The interaction between an initial 1h state and a 2h1p state essentially depends on their separation-energy and coupling strength. Thus if an initial 1h state approaches the double-ionization threshold with an increase of U, the interaction strength increases dramatically. When U becomes very large, the interaction becomes essentially that
between an initial 1h state and an unperturbed Coulomb-bound 2h1p state. Thus the initial core– hole spectral lineshape can depend critically on U. As an example, a many-body model calculation is performed to provide an overall description of the 4s- and 4p-hole XPS spectra of Pd to La [2,3]. We shed light on the mechanism of the anomalous spectral-lineshape change with the atomic number. In an interacting many-electron system, doubly subtracted electrons or two quasi-particles could be simply not a stable component of the system: They could and would decay into two or a number of constituent parts and the system’s true exact excitation spectrum is not a 2h-like at all. Moreover, the state is no longer a 2h bound-state because of the finite lifetime-effect. In general such a 2h state can be interpreted in terms of a resonance and 2h boundstates embedded in a 3h1p continuum. In fact it is entirely possible for a real, stable 2h bound-state to exist even when the coupling is strong. What inter-
0368-2048 / 01 / $ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 01 )00274-2
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
ests us most about a 2h bound state embedded in a 3h1p continuum is, not so much the possibility of its existence, as the fact that if it can be formed in the presence of strong coupling, then a fortiori it must be possible to produce a sharp resonance. In other words, the presence of a sharp resonance is not necessarily restricted to a regime of weak coupling. To demonstrate the many-body effect in a 2h state, we employ a many-body model calculation approach. We study the effect of the particle–hole interaction U in a 3h1p state on a 2h state. The density of 2h states depends critically on U. Thus the AES spectral-lineshape can be governed by U. As examples we consider the M 45 –N 23 N 45 AES spectra of Pd to Ba, which show anomalous lineshapechanges with the atomic number [4–7]. We provide an overall description of the density of final states changes with the atomic number. We consider also a similar particle–hole interaction effect on an AES spectrum of a charge transfer (CT) system.
2. Density of two-hole states
2.1. Theory We consider a 2h state (ij) in which one of the holes, i, involves a strong electron-correlation so that a 2h state interacts with 3h1p continuum states ( j 22 1 21 ´). Here we consider the ‘‘hole-hopping’’ relaxa–1 21 21 tion (screening), namely i –j 1 ´ in the presence of an extra j hole. The density of perturbed 2h states is 1 D rf (v ) 5 ] ? ]]]]]]]]] p (v 2 ´i 2 ´j 1 Uij 2 L)2 1 D 2
(1)
Here ´i (´j ) is the energy of hole i ( j). The bare hole–hole interaction, Uij, is included to infinite order by the ladder approximation. L and D provide the energy-shift and lifetime-broadening of a 2h state by the coupling with 3h1p states, respectively. L and D are F 2 U(F 2 1 D 2 ) L 5 2uV u 2 ]]]]]] (1 2 UF )2 1 (UD)2
(2)
D D 5 2uV u 2 ]]]]]] (1 2 UF )2 1 (UD)2
(3)
Here F(v ) and D(v ) are the real part and imaginary part of the unperturbed 3h1p propagator Go , respectively V is the energy-independent coupling matrix element between a 2h state and 3h1p states. The factor of two comes from the relaxation of hole i and the screening of Uij by the aforementioned ‘‘hole-hopping’’ relaxation. The unperturbed 3h1p propagator Go is
E
r (´) Go (v ) 5 ]]]]] d´ v 1 ´ 2 ´j 2l 2 id
(4)
´, v and ´j 2l are the particle-state energy, the 2h energy and the 3h energy, respectively. r is the density of unperturbed particle-states. Here unperturbed means without the screened particle–hole interaction U in the 3h1p states. The holes in the 2h and 3h1p states are assumed to be atomic-like localized so that the hole–hole interaction energy in the 3h1p states is taken into account by introducing the effective 3h energy. We obtain L and D by renormalization of 2 uV u 2 Go by including the screened particle–hole interaction U in the 3h1p state, to infinite order by the ladder and ring approximation. Here screened means the screening of the bare particle–hole interaction in terms of the particle–hole pair excitations. U is assumed to be particle-energy independent. We approximate the density of unperturbed particle-states by a parabolic band.
F S
3 ´ 2W r (´) 5 ] 1 2 ]] 4W W otherwise r (´) 5 0
D G for 0 # ´ # 2W, 2
(5)
r is normalized is 1.0. 2W is the width of density of the unperturbed particle-states. Introducing E 5 W 1 v 2 ´j 2 l we rewrite Eq. (1) 1 D rf (E) 5 ] ? ]]]]]] p (E 2 Eo 2 L)2 1 D 2
(6)
Here Eo 5 ´i 1 ´j 2 Uij 2 (´j 2 l 2 W ), is the unperturbed 2h state energy relative to the center of the 3h1p states (E50). Here unperturbed means without the coupling with the 3h1p states of interest. E 5 W corresponds to the 3h state energy v 5 ´j 2 l In the M 45 –N 23 N 45 Auger transition of Pd to Ba, the 4p4d (2h) state is coupled with the 4d 23 ´f (3h1p) states. The relativistic average-energies for
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
´i 1 ´j 2 Uij (4p 21 4d 21 2h state) and ´j 2l (4d 23 3h state) of metallic elements Ag to Ba are calculated from the relativistic atomic D SCF hole energies by using the atom–metal energy-shifts estimated by the method described in Ref. [8]. Thus the energy-shift of Eo by the ‘‘no-hole hopping’’ relaxation (screening) is already taken into account by the D SCF approximation. We consider explicitly the hole-hopping relaxation (screening) by taking into account the coupling between the 2h (4p4d) state and the 3h1p (4d 23 ´f) states. The AES spectrum is essentially a superposition of the density of the 2h states, separated by the initial core-level spin–orbit splitting, multiplied by the Auger-transition matrix element and statistically weighted by the ratio of the initial core–electron population. We emphasize that we are not aiming to describe the details of the experimental AES spectral lineshapes. We are more interested in the origin of
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the anomalous AES spectral-lineshape changes with the atomic number. Thus for the sake of clarity, we study the density of the 2h states rather than the AES spectral lineshape. Thus in Figs. 1 and 2 we present L, D, E 2 Eo and the density of 2h states (Eq. (6)), calculated for different U, Eo and V values (W53.0 is fixed). E is in units of Rydberg.
2.2. Results and discussion 2.2.1. M45 –N23 N45 Auger transition of Pd to Ba In Fig. 1 we show the density of 2h states with a change of U and Eo when V50.707. Fig. 1a (U 50 and Eo50) and Fig. 1b (U 50.3 and Eo50.5) correspond to Ag and In, respectively. U is small and Eo lies near the center of the 3h1p continuum states. There is a Es which satisfies Es 2 Eo 2 L (Es)50. However, dL(E) / dE at Es is smaller than 1.0 and is positive. The renormalization factor for the
Fig. 1. The density of two-hole states, rf , obtained by the following parameter values (V50.707 and W53.0 are fixed). (a) Ag: U 50.0. Eo50.0, (b) In: U 50.3, Eo50.5, (c) Sb: U 50.5, Eo51.3, (d) Te: U 50.7, Eo51.8, (e) I: U 51.4, Eo52.2, (f) Xe: U 52.1, Eo54.3, (g) Ba: U 52.5, Eo54.6. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Fig. 2. The density of two-hole states, rf , obtained by the following parameter values (V50.354 and W53.0 are fixed). (a) U 50.0, Eo50.0, (b) U 50.3, Eo50.5, (c) U 50.5, Eo51.3, (d) U 50.7, Eo51.8, (e) U 51.4, Eo52.2, (f) U 52.1, Eo54.3, (g) U 52.5, Eo54.6. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
density of 2h states, Z 5 (1 2 dL(E) / dE)21 , is larger than 1.0 at Es. Z provides the probability that the initial 2h state remains in the same state because 2 dL(E) / dE provides, to the lowest order in the perturbation theory, the probability that the initial 2h state makes a transition to a 3h1p state. When Z is negative or larger than 1.0, the ‘‘solution’’, Es,does not have any physical meaning and Z cannot be defined. To describe the density of 2h states by the Breit– Wigner resonance-lineshape, namely a resonant 2h state in 3h1p continuum states, a number of specific assumptions must be fulfilled. The most important of them is that the width and the shift from Es to Er (Er is the resonance energy) are both small relative to the energy scale on which the other quantities in the excitation amplitude vary appreciably. If the shift, Er 2 Es, is not small, the Taylor expansion around Es to obtain Z is no longer useful near Er. The shift
is given by 2 D(E)xZ 2 (dD(E) / dE) at Es. Thus if D(E) varies slowly (or is constant) and is not large, then Er 2 Es is small (or zero) and the Taylor expansion is valid. Otherwise, Er deviates substantially from Es. Z cannot be defined at all. If the width (D(E)) is not small, the resonance peak is superposed upon a background that varies as rapidly as the peak itself. Its shape may then be quite distorted; in fact, it may effectively become invisible. In the present case D(E) is rather constant around Es. Thus the shift, Er 2 Es, is small. However, D is large. Thus the conditions for the Breit–Wigner resonance-lineshape are not really satisfied. Thus it is appropriate to say that the notion of quasi-particle becomes illusory. Es is close to Eo. Thus the Auger-electron energy evaluated by the D SCF approximation (including the atom-metal energy shifts) becomes close to the center of the AES spectral lineshape. The relaxation
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
energy-shift of the 2h state, which is approximately Es 2 Eo, is small but the damping is large. The time scale of the hole decay is much shorter than that of the relaxation so that the relaxation becomes incomplete. The screening of the hole–hole interaction becomes incomplete because of absence of the holehopping screening. It should be noted that despite that one of the holes (4p hole) decays very rapidly, the hole–hole interaction in the 4p4d state does not become zero. Figs. 1c and 1d correspond to Sb and Te, respectively. With an increase of U, the density of the particle states in the 3h1p states begins to surge towards the triple-ionization threshold. The unperturbed 2h state begins to approach to the maximum of the density of the 3h1p states. The interaction between the unperturbed 2h state and the 3h1p states becomes large. D(E) is large and varies appreciably around Es. In Fig. 1c Er 2 Es becomes large. Thus the conditions for the Breit–Wigner resonancelineshape are not really satisfied for both cases. Thus it is appropriate to say that the notion of quasiparticle becomes illusory. Around Es, E 2 Eo and L becomes almost parallel. Then the density of 2h states becomes approximately 1 / hpDj. This explains a dip in the density in Figs. 1c and 1d. One may see the interaction between a 2h state with a 3h1p states as a kind of dissociative broadening. The dissociative broadening is the broadening of a localized 2h state in an alloy. The broadening arises from the mixing with delocalized band-states. In alloys a 2h bound state may mix with states with 1h in a host band and 1h in a partner-element band. The broadening becomes substantial when the host two-electron binding energy is in a large density of states region of ‘‘mixed convolution’’ of the host band with the partner band [9,10]. Fig. 1e corresponds to I. With an increase of U, the density of the particle states surges dramatically near the triple-ionization threshold. At the same time the unperturbed 2h state approaches the threshold. Thus the strength of the interaction between the unperturbed 2h state and the 3h1p states increases significantly. As a result of the strong interaction between the unperturbed 2h state and the 3h1p continuum states, an unperturbed 2h bound state splits off and appears below the threshold. The rest of original strength of the 2h state goes to the 3h1p
73
continuum states and the quasi-particle approximation breaks down. In a strong coupling regime a 2h bound state splits off as a sharp resonant state for which the quasiparticle approximation is valid. Es coincides with Er because D(E) 5 0. The relaxation energy shift Es(Er)2Eo is substantially large. The relaxation is complete because the decay time is much longer than the relaxation time. The relaxation energy-shift includes the relaxation energy-shift of a hole and the energy-shift of the hole–hole interaction energy by the ‘‘hole-hopping’’ screening. Figs. 1f and 1g correspond to Xe and Ba, respectively. With an increase of U, an unperturbed Coulomb-bound particle state splits off from the continuum and appears below the ionization threshold. Such a bound state dominates in the density of the particle states. As a result, the density of 2h states is essentially governed by the interaction between the 2h state and the 3h1p bound state. The unperturbed 2h state lies below the 3h1p bound state. Thus the perturbed 2h bound state (Es) which fulfills Es 2 Eo 2 L (Es) 5 0, is a Koopmans-theorem (KT)-like 2h state. The rest of the original strength of the 2h state goes to the 3h1p continuum. The quasi-particle approximation breaks down for a 2h state in the 3h1p continuum. If an unperturbed 2h state lies above an unperturbed Coulomb bound 3h1p state, then a bound state (Es) which fulfills Es 2 Eo 2 L (Es) 5 0 and D(Es)50. becomes a perturbed 3h1p state. The existence of an unperturbed 3h1p bound state is governed by the potential-binding condition 1 2 UF 5 0 and D 5 0, while that of a perturbed 3h1p bound state is governed by E 2 Eo 2 L(E) 5 0 and D(E)50. The former state is a split-off state which depends on the ratio of U to the continuum bandwidth, while the latter is a state which depends on the relative energy Eo. The mechanism for the appearance of the former state differs from that of the latter state. When the density of 3h1p states becomes a narrow and dense band with an increase of U, and it becomes degenerate and interacts strongly with an unperturbed 2h state, a perturbed 2h bound-state splits off (Fig. 1e). It is incorrect to interpret the split-off mechanism as that of the split-off of a localized 2h state when 2h are created in a narrow
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
band. A localized 2h state arises when the potentialbinding condition 1 1 UF 5 0 and D50 are fulfilled. Here F and D are the real part and imaginary part of the two-hole propagator without the hole–hole interaction U, respectively. U is the hole–hole interaction energy. The split-off of a perturbed 2h bound state from the band depends on the relative energy Eo and the coupling strength V, namely on the interaction between two different states, not on the potentialbinding condition, while the split-off of an unperturbed 3h1p bound state from the band (Figs. 1f and 1g) depends on the potential-binding condition and is quite analogous to that of the appearance of a localized 2h state in the C2VV AES spectrum of a narrow-band metal. The potential-binding condition is governed by the particle-hole interaction in a 3h1p state. The present many-body calculation reveals the mechanism of anomalous spectral-lineshape changes of the M 45 –N 23 N 45 AES spectra of Pd to Ba with the atomic number. With U increase, the density of particle states in the 3h1p states surges towards the triple-ionization threshold. The density of particle states becomes narrow and dense. When U becomes
large, a Coulomb-bound particle state splits off from the continuum and appears below the threshold. Such a bound state dominates in the density of particle states. U increases with the atomic number. At the same time an unperturbed 2h state approaches to the triple-ionization threshold (see Fig. 3). As a result the interaction between an unperturbed 2h state with a 3h1p continuum and / or bound state increases. It is U which increases predominantly the coupling strength between a 2h state and 3h1p states with the atomic number. This most important point was not realized in a theoretical study of the M 45 –N 23 N 45 AES spectra of Pd to Xe by the ab-initio atomic Green’s function calculation by the author [4–6]. The present model calculation has an advantage that one can monitor the variations of the strength of physical quantities which govern predominantly the spectral-lineshape change. The major problem neglected so far is the multiplet structure of the 2h (4p4d) state. The present calculation shows that except for the case of a 2h bound state, the quasi-particle approximation for a 2h state breaks down. In such a case the concept of the multiplet splitting becomes illusory because the
Fig. 3. Theoretical relativistic atomic D SCF energies of single- and multiple-hole states of the elements Z546 to Z557.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
decay width is much larger than the multiplet splitting. This is related to the fact that delocalized holes do not form multiplet-coupled states. One may explain in an qualitative manner the density of 2h states as the 4p hole spectral-function of the Z 1l atomic element. The 4p hole in the presence of a 4d hole may be interpreted as that of the Z 1 1 atomic element. The density of the 3h1p states becomes rather similar to that of the 2h1p states of the Z 1 1 atomic element because of an increased U by the presence of an extra hole. The parameter value U for the 3h1p state is indeed larger than that for the 2h1p state. Moreover, it is closer to U for the 2h1p state of the Z 1 1 atomic element. However, the separation energy between the 2h state and the 3h1p states differs from that between the 1h state and 2h1p states of the Z 1 1 atomic element because of an increased hole–hole interaction in the 3h1p state by the presence of an extra 4d hole. Thus strictly speaking the Z 1 1 approximation is not valid. One cannot simply apply the Z 1 1 approximation for a transition which involves multipleholes. So far we considered the case when one hole in a 2h state (i ± j) induces a strong electron-correlation. We can treat also the case when both holes in a 2h state involve strong electron-correlations. When i 5 j, in general the factor of two in Eqs. (1)–(4) becomes four. However, when i 5 j 5 4p, the hole-hopping CK process of each 4p hole occurs, but the dipolar screening of the hole–hole interaction by the CK process does not occur. Thus the factor of two remains in Eqs. (1)–(4), One cannot simply approximate the density of 4p 22 (2h) states by a self-convolution of the 4p spectral function of the Z 1 1 atomic element because of the presence of the hole–hole interaction and many-body effects described above. So far there is no experimental study of the M 45 –N x N y (X5Y5 4s, 4p) AES spectra of Pd to Ba. In the M 45 –N 1 N 23 Auger transition, the 4s 21 4p 21 (2h) state interacts with the 4p 22 4d 21 ´f (3h1p) states. The 4p 22 4d 21 (3h) state interacts with the 4p 21 4d 23 ´f (4h1p) states. The cascade interaction continues. Thus the many-body effect becomes much more complicated than that of the M 45 –N 23 N 45 Auger transition. It would be interesting to investigate the M 45 –N x N y AES spectra of Pd to Ba by Auger-
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photoelectron coincidence spectroscopy (APECS) because the density of final 2h states can be directly obtained when the Auger electron is measured in coincidence with a selected initial core–hole state. The 4s XPS spectrum looks quite normal, except for a large relaxation energy shift from the D SCF hole energy, caused by 4s 21 –4p 21 4d 21 ´f CK virtual process [2]. When U is very small, the imaginary part of the 4s-hole self-energy is given by the convolution of the density of unperturbed particle states (´f ) with the density of the perturbed 2h (4p4d) states, multipled by the energy-independent CK transition matrix element. With an increase of U, the convoluted density will be perturbed by U. In general, the lifetime broadening of 2h states is negligible, compared to the density of particle states. Then the density of 2h1p states becomes that of particle states perturbed in the presence of two holes. This is the case with the 4p-hole self-energy. However, this is not the case with the 4s-hole self-energy. The density of perturbed 2h (4p4d) states is comparable with the density of particle states (´f ). Then the density of unperturbed 2h(4p4d)1p(´f ) states, given by a convolution of the density of unperturbed particle states (´f ) with the density of perturbed 2h (4p4d) states, is much broad, compared to the density of unperturbed particle states. Both the relaxation energy-shift and lifetime broadening of a 4s hole can be affected by the broadening of the density of 2h(4p4d)1p(´f ) states. The 4s hole involves a much more complicated many-body effect than a 4p hole, despite that the former hole spectrum looks much more normal than the latter one. The many-body effect does not manifest in the ionization spectral-lineshape.
2.2.2. V50.354 case Eo lies in the 3h1p continuum (Figs. 2a–2d). However, because of a small coupling strength, D becomes small and rather constant around Es so that the energy shift, Er–Es, becomes small. As a result the density of 2h states becomes a resonant-like lineshape. It is appropriate to say that the quasiparticle approximation for the 2h state is valid. When Eo lies below a Coulomb-bound 3h1p state (Figs. 2f and 2g), the relaxation energy-shift of the perturbed 2h bound state becomes smaller and its spectral
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
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intensity becomes larger because of a smaller coupling strength. The one-electron picture becomes valid.
3.1. Theory For the sake of simplicity we consider the case when two localized-holes are created in the same inner-valence level in an acceptor. The density of the perturbed 2h states is (7)
Here v0 is the unperturbed 2h state energy without the CT interaction. They are F 1 U(F 2 1 D 2 ) L 5 4uV u 2 ]]]]]] (1 1 UF )2 1 (UD)2
(8)
D D 5 4uV u 2 ]]]]]] (1 1 UF )2 1 (UD)2
(9)
V is the energy-independent coupling matrix element. F and D are the real part and the imaginary part of the unperturbed 3h1p propagator, respectively. The unperturbed 3h1p propagator is
r (´) E]]]]]] d´ 5 2 F 1 iD ´ 1 v 2 ´ 2 v 2 id a
d
D G for 0 # ´ # 2W, 2
(11)
Then we introduce E 5 vo 2 v 2 W 1 ´d and Eo 5 ´d 2 W. From Eq. (7) we obtain the density of perturbed states two-hole.
3. Charge transfer system
1 D r (v ) 5 ] ? ]]]]]] p (v 2 v o 2 L)2 1 D 2
F S
3 ´ 2W ra (´) 5 ] 1 2 ]] 4W W otherwise ra (´) 5 0
(10)
o
Here unperturbed means without the particle–hole interaction in the 3h1p CT state. We renormalize the unperturbed 3h1p propagator (Eq. (10)) by including the screened particle–hole interaction of U, to infinite order by the ladder and ring (low density) approximation. Then multiplying by uV u 2 , we obtain the real part (Eq. (8)) and the imaginary part (Eq. (9)) of the renormalized term which provides the CT interaction between the 2h state and the 3h1p CT states. The factor of four in Eqs.8 and 9 comes from the screening of each inner-valence hole and that of the bare hole–hole interaction between two innervalence holes. We approximate ra by a parabolic band given by
1 D r (E) 5 ] ? ]]]]]] p (E 2 Eo 2 L)2 1 D 2
(12)
E 5 Eo corresponds to v 5 vo , while E 5 W(2W ) does to v 5 ´d 1 vo 2 2W (v 5 ´d 1 vo ), namely the unperturbed 3h1p state in which there is CT from the donor to the top (bottom) of the unoccupied acceptor-band in the presence of two valence-holes. For the density of perturbed 2h states, the energy scale E is the negative of 2h energy. r obtained for different U and Eo are shown in Figs. 4 and 5, together with L, D and E 2 Eo.
3.2. Results and discussion Figs. 4a–4d correspond to the cases when there is an excitation predominantly from below the Fermi level. When U is small (Figs. 4a and 4b), the smallest binding energy state is a Koopmans theorem (KT)-like two-hole state, in which a screening electron still remains predominantly in the occupied part, i.e. there is almost no extra screening. The density of KT-like 2h state increases with a decrease of V because of a smaller coupling strength. The continuum state is a shakeup state, in which an electron is excited from the occupied part to the unoccupied part of the resonance density of states. The quasi-particle approximation breaks down for the satellite. With an increase of U (Figs. 4c and 4c), the density of the unoccupied states surges toward the Fermi level. When U becomes larger than the width of the unoccupied part of the resonance density of states, an empty bound-level splits off from the band states and appears below the Fermi level. When an empty bound-level in the acceptor still lies just below the Fermi level (Fig. 4c), the smallest binding energy state is a KT-like state, in which a screening electron still remains predominantly in the occupied part, i.e., no extra hole screening. The density of a
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Fig. 4. The density of the two-hole states, r, obtained by the following parameter values. W 5 1.0, V 5 0.5, Eo5 22.0, (a) U 5 0.0, (b) U 50.3, (c) U 50.8. (d) U 52.0. Also shown are L (solid line), D (dotted line) and E 2 Eo (straight line).
KT-like state increases with a decrease of V because of a smaller coupling strength. In Fig. 4c the CT interaction between the occupied part and the empty bound-level in the acceptor is still not large enough so that the satellite is still a shakeup state, in which an electron is excited from the occupied part to the unoccupied part of the resonance density of states. The quasi-particle approximation becomes valid for the satellite. When U becomes larger (Fig. 4d), an empty bound-level in the acceptor appears predominantly below the occupied part. Then the smallest binding energy state is a CT shakedown state, in which the empty bound-level in the acceptor is predominantly occupied by an electron from the occupied part. There is an extra screening. There is a KT-like state in which a screening electron still remains pre-
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Fig. 5. Same as Fig. 4, except for Eo5 21.0.
dominantly in the occupied part. It is a bound state, the lineshape width of which is due to the valencehole lifetime. With a decrease of U and V, the density of the KT-like state increases. There are shakeup states of non-negligible intensity. Figs. 5a–5d correspond to the cases when there is an excitation predominantly from the Fermi level. When U is small (Figs. 5a and 5b), the behavior of DOS is still similar to that in Fig. 4a and 4b. When U is large (Figs. 5c and 5d), the KT-like state becomes a resonant state. The lineshape-width is nothing but the effective excitation rate at the resonance energy (except for the valence-hole lifetime-broadening). Thus the effective CT coupling strength at the resonance can be determined from the lineshape of the density of 2h states. There is also a shakedown state, in which an electron is transfered from the occupied part to an empty bound-level for the hole screening. With a decrease of V, the density of the KT-like state increases. With an increase of U, the
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
density of the KT-like state decreases. However, when U becomes comparable with the bandwidth of the unoccupied part of the resonance density of states, an empty bound-level splits off from the band. With an increase of U, the density of the KT-like resonant state increases. The major difference between the AES spectra discussed in Section 2.2 and the present one is, that in the latter the unperturbed 2h state always lies below or at the edge of the 3h1p states. The case discussed for Fig. 1f is very similar to that of Fig. 4c. The case when a 3h1p bound state lies below an unperturbed 2h state discussed in Section 2.2 is found in Fig. 4d in which the perturbed 3h1p bound state lies below the perturbed 2h state. The case shown in Fig. 2e is close to that in Fig. 5c, In the former case, because of small ratio of (U /W )50.47 compared to that (0.8) in the latter case, there is no unperturbed 3h1p bound state below the band so that a 3h1p bound state does not split off, while in Fig. 5c there is an unperturbed 3h1p bound state so that a 3h1p bound state splits off. Both Figs. show the presence of a resonant KT-like 2h state. The width of the resonant-like state spectral-lineshape is governed by the coupling strength between the 2h state and the 3h1p state. Thus it is self-explanatory that the same mechanism is governing both spectra.
4. Conclusion By using a many-body model calculation, we study the effect of the particle-hole interaction, U, in 3h1p states on a 2h state. With an increase of U, the density of particle states in the 3h1p states will be pushed towards the triple ionization threshold. The density of particle states becomes narrow and dense. If an unperturbed 2h state shifts towards the threshold at the same time, the interaction between an unperturbed 2h state and perturbed 3h1p states increases dramatically so that a 2h bound state splits off from the band and appears below the threshold. A sharp resonant 2h state appears even in a strong coupling regime, while the quasi-particle approximation may break down in a ‘‘weak’’ coupling regime. When U becomes large, a Coulomb-bound particle state splits off and appears below the threshold. It
takes a large part of the density of particle states. The interaction between a 2h state and 3h1p states becomes essentially that between a 2h state and a 3h1p bound state. As a result, the density of 2h states dominantly consists of discrete states. The present model calculation can provide an overall description of variations of the density of 2h states of the M 45 –N 23 N 45 Auger transition of the elements Pd to Ba. U increases with the atomic number and causes an anomalous AES spectral behavior. The many-body effects associated with the ionization of 4s- and (or) 4p- electrons are still not fully investigated, despite that they provide very unique spectral behaviors. In particular, the cascade decay of 4p(4s) hole in a multiple-hole state may provide unique opportunities to study the post-collision interaction effect on an Auger-electron spectrum, because of a sudden change in the Coulomb potential a slow photoelectron (an Auger-electron) experiences. Further experimental studies using Auger-electron spectroscopy and / or Auger-photoelectron coincidence spectroscopy are needed. The effect of particle–hole interaction on an AES spectrum of a CT system is discussed. When the interaction of an electron transfered from the occupied part to the unoccupied part of the resonance density of states perturbed by two inner-valence holes created by core–hole decay, becomes larger than the bandwidth of the unoccupied part of the resonance density states, an empty acceptor boundlevel splits off from the band. This is governed by the potential-binding condition. When such an empty acceptor bound-level is pulled below the occupied part, a shakedown state appears as a screened two-hole state. When there is an excitation predominantly from the occupied part close to or at the Fermi level, there is a possibility that a KT-like 2h state appears as a resonant state, the width of which is determined by the sum of the excitation rate at the resonance and the valence-hole lifetime-broadening. In such a case one can monitor the excitation rate change with a chemical environmental change. The mechanism which governs the AES spectral-lineshape of a CT system considered here is the same as that proposed for the M 45 –N 23 N 45 AES spectral-lineshape changes with the atomic number. So far there appears to be no relevant experimental observation for a CT system. However,
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 119 (2001) 69 – 79
Auger-photoelectron coincidence spectroscopy (APECS) will be useful because it reveals directly the density of final 2h states.
References [1] M. Ohno, J. Elect. Spectr. Rel. Phenom. 106 (2000) 37. [2] M. Ohno, J. Elect. Spectr. Rel. Phenom. 107 (2000) 113. [3] M. Ohno, J. Elect. Spectr. Rel. Phenom. 119 (2001) 19–27.
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[4] M. Ohno, G. Wendin, Solid State Commun. 39 (1981) 875, Energies plotted in the spectra of this reference and Ref. [5] The SCF are not accurate. [5] M. Ohno, J.-M. Mariot, J. Phys. C14 (1981) L1133. [6] M. Ohno, Phys. Rev. A38 (1988) 3473. [7] S. Aksela, H. Aksela, T.D. Thomas, Phys. Rev. A19 (1979) 721. [8] M. Ohno, G. Wendin, J. Phys. C15 (1982) 1787. [9] P.A. Bennett, J.C. Fuggle, F.U. Hillebrecht, A. Lenselink, G.A. Sawatzky, Phys. Rev. B27 (1983) 2194. [10] M. Ohno, J. Elect. Spectr. Rel. Phenom. 107 (2000) 103.