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On the satellite in the Auger electron spectrum
Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29–36 www.elsevier.nl / locate / elspec
On the satellite in the Auger electron spectrum Masahide Ohno* Advanced Physics Research Laboratory, 2 -8 -5 Tokiwadai, Itabashi-ku, Tokyo, 174 -0071, Japan Received 4 March 1999; received in revised form 11 May 1999; accepted 11 May 1999
Abstract The satellite line profile in the C–VV Auger electron spectroscopy spectrum depends upon how the initial core hole state and the final multiple hole state are created. Even if the numbers of the holes in the satellite states are the same, the Auger electron spectroscopy spectral profiles of the final multiple hole states are different. The localization of the three hole satellite state is discussed. 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Photoionization is often a very rapid ‘instantaneous’ application of the perturbation (i.e. in a short time compared with the periods of transition from a given state to other state). In contrast to the adiabatic limit where the entropy of the system remains unchanged, i.e. the process is reversible, the sudden limit is the nonadiabatic limit where the process is irreversible. The nonadiabatic ionization process induces a number of correlated ionized states, because of a conservation of oscillator strength, which requires that any suppression of the direct transition be compensated for by the matrix element to other excited states of the system, in which additional electrons are excited from their unperturbed states. Friedel [1] called such additional excitations the shakeoff. The shakeup / off satellites in the core (valence) electron ionization spectra have been studied to elucidate the ground state electronic structures. However, the satellite observed in the Auger electron *Corresponding author.
spectroscopy (AES) spectrum has not been of much concern, except in the case of C–VV AES spectra of the elements around Cu (free atoms, metals, compounds) where the final three valence-hole state can be created by different decay channels. These AES spectra have been the subject of intensive study [2–8]. If the initial core shakeoff satellite state does not relax to the initial core main-line state on the time-scale of core-hole decay, both states can decay independently. Then, the initial core shakeoff satellite state can decay to the three valence-hole state (CV–VVV), while the final state valence shakeoff upon Auger decay of the initial main-line state results in the three valence-hole state too (C–VVV). Coster–Kronig (CK) decay preceded Auger decay results also in the three valence-hole state (C–CV– VVV). Recently, Auger photoelectron coincidence spectroscopy (APECS) has begun to show its potential for studies of electron correlations [8–11]. Using APECS, one can determine the correlations between the initial state ionizations and the Auger final states by measuring the photoelectron (or Auger electron) in coincidence with the specified Auger final state (or
0368-2048 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 99 )00032-8
30
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
specified initial core-hole ionized state). A recent APECS study of C–VV Auger transition of the 3d transition metals and related systems showed the importance of the decay of the initial shakeup / off satellite state to the three hole-state [9,11]. Thurgate [9] stated that it should be recognized that there is no reason that the initial state and final state shakeup / off should produce the same effect on the Auger line shape; the initial state shakeoff produces a spectator hole that may influence the valence band structure and interact with the out-going Auger electron, while the final state shakeoff produces an extra hole during the Auger electron emission process. The final state may be the same, but the initial state is different. The AES spectral profile is essentially described by the convolution of the initial core-hole spectral function and the final state spectral function (density of the final states), weighted by the partial decay rate ratio [12]. Thus, one should recognize that the AES line profile must be different when the initial state and the final state are created by different processes. In this paper, it will be theoretically shown that the Auger satellite line profile depends on how it is created. The localization of the three-hole satellite state will be discussed. We take into account the screening of the bare hole hole interaction and the relaxation (fluctuations) and decay of hole(s) by renormalizing the multiple-hole Green’s function.
one-particle shakeoff state. For the sake of simplicity, we neglect the multiplet structure of the final state. The valence-hole spectral function is given by Im SV (E) 1 A V (E) 5 ] ]]]]]]]]]] p (E 2 ´V 2 Re SV (E))2 1 (Im S V (E))2 (2) We consider the two-hole Green’s function by renormalizing each hole’s propagator by the selfenergy in the presence of an extra hole and screening the bare hole-hole interaction by the hole-particle pair excitations. Then, the two-hole Green’s function for the holes i and j (i ±j) becomes Gij (E) 5sE 2 ´i 2 ´j 1 Uij 2 Si (E) 2 Sj (E) 2 Sij (E) 2 Sji (E)d 21
(3)
Here
Si (E) 2
5S
l,m
uV (´)u E ]]]]]]]]]]] d´ E 2´ 2´ 2´ 1U 1U 1U 1´ iml ´
l
m
j
lm
mj
lj
(4)
Sj (E) 5S
l,m
E
uVjml ´ (´)u 2 ]]]]]]]]]]] d´ E 2 ´l 2 ´m 2 ´i 1 Ulm 1 Umi 1 Uli 1 ´ (5)
2. AES spectral profiles We consider the case where the dominant part of the satellite is due to the shakeoff process and the valence hole(s) is localized. In Section 4, we extend the formalism to the case where the hole(s) is created in the valence band. For the valence electron ionization, the valence-hole self-energy is given by
Sij (E) 5S l
1S l
uV u r (´) E ]]]]]] d´ E 1 ´ 2 2´ 1 U V
s
V
(1)
VV
Here ´V is the Koopmans valence hole energy, ´ is the shakeoff electron energy, rs is the density of the unoccupied states for the shakeoff electron. UVV is bare valence hole-hole interaction and Vv is the energy-independent coupling matrix element between the single valence-hole state and the two hole
uV (´)V (´)u E ]]]]]]]]]] d´ E 1´ 2´ 2´ 2´ 1U 1U 1U ili ´
i
Sji (E) 5S l
2
SV (E) 5
1S l
E
uVilj ´ (´)u 2 ]]]]]]]] d´ E 1 ´ 2 ´l 2 2´j 1 2Ulj 1 Ujj
E
l
jlj ´
j
il
lj
(6)
ij
uVjli ´ (´)u 2 ]]]]]]]] d´ E 1 ´ 2 ´l 2 2´i 1 2Uli 1 Uil
uV (´)V (´)u E ]]]]]]]]]] d´ E 1´ 2´ 2´ 2´ 1U 1U 1U jlj ´
i
l
ili ´
j
il
lj
(7)
ij
Here the ‘large S’ denotes the summation over the hole indices. Eq. (4) describes the relaxation and decay of hole i( j) in the presence of an extra hole j(i). Eqs. (6) and (7) provide the screening of the bare hole-hole interaction between i and j by the hole particle pair excitations. V is the Coloumb
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
matrix element and includes the exchange term implicitly. U is the bare hole-hole interaction, included to infinite order by the ladder approximation. Eqs. (4)2(7) include the relaxation (decay) of the holes i and j or the screening of the bare hole-hole interaction between i and j to second order. The bare hole-hole interaction among the final state holes is included to infinite order, but the relaxation (decay) of the holes and the screening of the bare hole-hole interaction are neglected. One can include the relaxation of the final state holes and the screening of the bare hole-hole interaction by introducing the threehole (one particle) propagator and repeat the same renormalization procedure. However, as a simpler approximation for the relaxation and decay of the final state holes, one may replace the Koopmans hole energies in the denominators of Eqs. (4)–(7) by the renormalized hole energies. El 5 ´l 1 Dl (El ) 1 iGl (El )
(8)
31
A V 2 (E) 5 2 Im SV (E) 1 ] ]]]]]]]]]]]]]] p (E 2 2´V 1 UVV 2 4 Re SV (E))2 1 (2 Im SV (E))2 (11) Here SV (E) is the valence-hole (shakeoff) selfenergy given by 2
SV (E) 5
uV u r (´) E ]]]]]]]] d´ E 1 ´ 2 3E 1 3U 2 6D V
s
V
VV
(12)
V
In the same manner, one can obtain the three-hole spectral function (the density of three-hole states). A V 3 (E) 5 3 Im SV (E) 1 ] ]]]]]]]]]]]]]] p (E 2 3´V 1 3UVV 2 9 Re SV (E))2 1 (3 Im SV (E))2 (13)
Here D and G are the real and imaginary part of the hole self-energy given by
Sl (E) 5n,m S
E
uVlnm ´ (´)u 2 ]]]]]]] d´ E 1 ´ 2 ´n 2 ´m 1 Unm
(9)
El is the solution of the Dyson equation given by El 5 ´l 1 Re Sl (El )
(10)
For the screening of the bare hole-hole interaction in the denominator of Eqs. (4)–(7), one can replace the bare hole-hole interaction, Uij , approximately by Uij 2 Di 2 Dj . Here, D is given by Eq. (8). We neglect the effect of the presence of an extra hole. When i 5 j, the first term and the second term of Eq. (6) become identical. When i 5 j, the imaginary part of the self-energy given by Eq. (6) becomes identical to that given by Eq. (4) from a viewpoint of unitary expansion. Thus, when i5j, the imaginary part of the self-energy by Eqs. (6) and (7) does not contribute. The decay width of the two-hole is approximately given by twice the single-hole decay width. When the two holes are created in the same valence shell (v), the two-hole spectral function (density of twohole states) becomes
So far, the interaction between the particle (shakeup / off electron) and the holes in the final state has been neglected. The interaction, U, can be included to infinite order by the ladder and ring approximation. By defining the real and imaginary part of the selfenergy given by Eq. (12) as follows
SV (E) 5 F 1 iD
(14)
The real and imaginary part of self-energy renormalized by including U to infinite order by the ladder and ring approximation become F 2 U(F 2 1 D 2 ) Re SV (E) 5 ]]]]]] (1 2 UF )2 1 (UD)2
(15)
D Im SV (E) 5 ]]]]]] (1 2 UF )2 1 (UD)2
(16)
We now consider the shakeoff upon the core electron ionization. The unrenormalized core-hole self-energy for the shakeoff is given by 2
S sc (E) 5
uV u r (´ ) E ]]]]]]] d´ E 1´ 2´ 2´ 1U c
s
s
C
s
V
CV
s
(17)
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
32
Here ´C , ´V , UCV and ´s are the Koopmans core-hole energy, the Koopmans valence-hole energy, the bare core hole–valence hole interaction and the shakeoff electron energy, respectively. rs (´s ) is the density of the unoccupied states for the shakeoff electron and Vc is the energy-independent coupling matrix element between the single core-hole state and the two-hole one-particle shakeoff state. When we take into account the decay and relaxation of the two-hole and the screening of the bare hole-hole interaction, Eq. (17) becomes
S sc (E) 5
E uV u r (´ )G 2
I mAES (´A )v 5
F
E A (´ 2 v) m
p uVCVV´A (´A )u 2 ]]]] A 2 (´ 1 ´ 2 v ) Im S Ac (´ 2 v ) V A
G
d´
(22)
Here, ´ and v are the photoelectron energy and the incident photon energy, respectively. A m is the mainline state part of the initial core-hole spectral function given by 1 A m (´ 2 v ) 5 ] p
(18)
Im S Ac (´ 2 v ) ]]]]]]]]]]]]] A (´ 2 v 2 ´c 2 Re S c (´ 2 v ) 2 Re S sc (´ 2 v ))2 1 (Im S Ac (´ 2 v ))2
Here Gcv is the one core-hole one valence-hole Green’s function (Eq. (3) for i 5 c and j 5 v). We consider also the Auger decay c 21 2 l 21 m 21 ´A of the initial core-hole (c). The corresponding unrenormalized self-energy of the core-hole is given by
(23)
C
s
s
CV
(E 1 ´s ) d´s
uVclm ´A (´A )u 2 S (E) 5 S ]]]]]]] d´A l,m E 1 ´A 2 ´l 2 ´m 1 Ulm A c
(19)
l,m
E uV
clm ´A
(´A )u 2 Glm (E 1 ´A ) d´A
(20)
Here Glm is the two-hole Green’s function (i 5 l, j 5 m in Eq. (3)). The core-hole Green’s function is given by Gc (E) 5 [E 2 ´c 2 S Ac (E) 2 S sc (E)] 21
Im S cA (´ 2 v ) 5 S
l,m
Here Vclm ´A is the energy-dependent Auger decay matrix element. When we take into account the decay and relaxation of two holes, l and m, and the screening of the bare hole-hole interaction, Eq. (19) becomes
S Ac (E) 5 S
The line profile of A m is broadened by the Auger decay of the core-hole. A v 2 is the final two valencehole spectral function (Eq. (11)). The imaginary part of the core-hole’s self-energy is given by (see Eq. (20))
(21)
We consider the case where two initial core-hole states (main line and shakeoff satellite) can decay independently. Moreover, the couplings to the same final state from different initial core-hole states are so small that the interference terms are negligible. The AES spectral line profile for the decay of the main-line state is given by
E p uV
clm ´A
2 v ) d´A
(´A )u 2 A lm (´A 1 ´ (24)
The AES spectral profile is given by the convolution of the initial state spectral function and the final state spectral function, weighted by the unrenormalized partial decay rate ratio. Here, the unrenormalized means the neglect of the final two hole state lifetime and interaction. When the AES spectrum is integrated with the Auger electron energy, the factor inside the parentheses becomes the renormalized partial decay rate ratio. The renormalized partial decay rate is the convolution of the unrenormalized partial decay rate with the two-hole spectral function. As the width of the two-hole spectral function is negligible compared to the ‘width’ of the density of the unoccupied states for the Auger electron, the effect is small. However, the CK decay electron energy is much smaller compared to the Auger electron energy. Moreover, the ‘width’ of the unoccupied states for the CK electron is much smaller than that for the Auger electron. Thus, the broadening may affect the decay rate near the double ionization threshold. When all decay channels are
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
considered, the total integrated ‘decay’ spectral intensity becomes the initial main line state spectral intensity. Here, ‘decay’ means both radiationless and radiation decays. For the Auger decay of the initial shakeoff satellite state, the AES spectrum becomes AES
Is
(´A )v 5
A (´ 2 v ) E ]]]] Im S (´ 2 v ) SE p uV u r (´ )A s
s c
2
c
F
s
s
cv
(´s 1 ´ 2 v )
˜ ´ )u 2 A 3 (´ 1 ´ 1 ´ 2 v ) p uV( A V s A ]]]]]]]]]]] ˜ Im Sc (´s 1 ´ 2 v ) 1 Im S˜ V (´s 1 ´ 2 v )
G D
d´s d´ (25)
Here A s is the satellite part of the initial core-hole spectral function given by 1 A s (´ 2 v ) 5 ] p Im S sc (´ 2 v ) ]]]]]]]]]]]]] A (´ 2 v 2 ´c 2 Re S c (´ 2 v ) 2 Re S sc (´ 2 v ))2 1 (Im S 2c (´ 2 v ))2
Im S˜ c (´s 1 ´ 2 v ) 5 S
l,m
E puV˜
clm ´A
33
(´A )u 2 A lmv (´s
1 ´A 1 ´ 2 v ) d´A
(28)
V˜ is the Auger decay matrix element for the core hole in the presence of a spectator valence hole. Thus, the self-energy S˜ c differs from the self-energy introduced by Eq. (19). The factor inside the parentheses becomes the renormalized partial decay rate ratio when the AES spectrum is integrated with the Auger electron energy. When all decay channels, including the decay of the valence hole, are considered, the factor inside the bracket becomes the imaginary part of the shakeoff self-energy (Eq. (27)). Then the total integrated decay spectral intensity becomes the integrated initial shakeoff satellite intensity. Compared to the AES spectral profile for the decay of the initial main-line state, the one from the initial shakeoff satellite state is much more complicated. So far, in the core hole self-energy, the particle-hole(s) interaction is neglected for the sake of simplicity. However, it can be included, as in Eqs. 15 and 16.
(26) The imaginary part of the shakeoff self-energy S sc (´ 2 v ) is given by (Eq. (18)) Im S sc (´ 2 v ) 5
E p uV u r (´ )A 2
C
s
s
CV
(´s 1 ´ 2 v ) d´s
3. Three hole state in the AES spectrum To reach the final three-hole state, there are two possible channels.
(27) A CV is the one core hole–one valence hole pair spectral function given by the imaginary part of GCV divided by p (i 5 c and j 5 v in Eq. (3)). V˜ is the C–VV Auger decay matrix element in the presence of a spectator valence hole. Note that the imaginary part of the shakeoff self-energy is given by the convolution of the density of the unoccupied states for the shakeoff electron and the core hole–valence hole pair spectral function A CV . Thus, the density of the unoccupied states is broadened by the lifetime of the core hole–valence hole pair. In general, the width of the former is much larger than the latter. Thus, the broadening effect is negligible. The imaginary part of the core hole self-energy S˜ c is given by
Channnel A: The final state shakeoff upon the Auger decay of the initial main line state, which is predominantly single core-hole state. Channel B: The Auger decay of the initial shakeoff satellite state, which is predominantly one core-hole and one valence hole state. For channel A, the AES spectrum is given by the satellite part of the AES spectrum given by Eq. (22). For channel B, the AES spectrum is given by the main line part of the AES spectrum given by Eq. (25). Thus, the AES spectral profiles for channels A and B are different not only in the Auger energies but also in the spectral profiles.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
34
For L 2 –L 3 V–VVVCK, decay preceded Auger decay, Eq. (25) becomes I AES L 2 – L 3 V–VVV (´A )v 5
SE
F
EA
L2
1 (´ 2 v ) ]]]] Im SL 2 (´ 2 v )
puVCK ( ´˜ )u 2 A L 3 V ( ´˜ 1 ´ 2 v )
puVA (´A )u 2 A v 3 ( ´˜ 1 ´A 1 ´ 2 v ) ]]]]]]]]]]] Im S˜ L 3 ( ´˜ 1 ´ 2 v ) 1 Im S˜ V ( ´˜ 1 ´ 2 v )
GD
d´˜ d´ (29)
A L 2 is the main line state part of the L 2 core-hole spectral function. Im SL 2 is the imaginary part of the L 2 core-hole self-energy. A L 3 V is the L 3 V two-hole spectral function. VCK is the energy-dependent L 2 – L 3 VCK decay matrix element and ´˜ is the corresponding CK decay electron energy. VA is the L 3 V– VVV Auger decay matrix element. S˜ L 3 is the selfenergy for the L 3 core-hole in the presence of a valence hole. If we compare Eq. (29) with Eq. (25) for the case of L 3 V (shakeoff)–VVV Auger decay, the following factors are different; the initial corehole spectral function, the core-hole self-energy and the shakeoff matrix element. Then the AES spectral line profile for the three-hole states by L 2 –L 3 V– VVV decay is different from that by the L 3 V–VVV decay. The APECS spectrum measured in coincidence with a particular initial core-hole state shows the spectral function of the final state reached by the Auger decay of the selected initial state, while the APECS spectrum measured in coincidence with a particular Auger final state shows the spectral function of the initial state for the Auger decay (the spectral line width will become narrower when the final state spectral width is narrower than the initial state one). The APECS can be used to separate the initial and final state events. However, one cannot obtain separately the AES spectral profiles for channels A and B. For channel A, the APECS spectrum essentially shows the final two-hole spectral function. If the final satellite state is well separated from the final main line state, one can obtain the main line part and the satellite part of the final state two-hole spectral function. For channel B, the situation is different.
The APECS spectrum shows essentially the final three-hole spectral function convoluted by the one core-hole one valence-hole shakeoff state spectral function. If the presence of multiplet levels in the initial shakeoff state is substantial, the APECS spectrum can be quite complicated. In that case, it is necessary to scan each initial shakeoff multiplet state so that one can decompose the APECS spectrum by the initial state events.
4. Localization of the multiple hole state So far, the valence holes are assumed to be localized. We consider the cases when holes are created in a valence band. The unperturbed two valence-hole Green’s function (‘unperturbed’ means before the valence hole-hole interaction is switched on) is given by GVV (E) 5
r (´) E ]]]]] d´ E 2 ´ 2 2S (E) V2
(30)
V
Here, rV 2 (´) is the self-convolution of the Koopmans density of the valence hole states given by
rV 2 (´) 5
E r (´ 2 ´9)r (´9) d´9 V
V
(31)
´ is the sum of two Koopmans valence-hole energies. The relaxation and decay of two holes could be included by the hole self-energy SV (E). In Eq. (30), the relaxation effect on the density of the hole states is neglected by using the Koopmans density of the states (Eq. (31)). By using the effective density of the hole states, r˜ V , which corresponds to the valence hole spectral function (main line state), instead of rV in Eq. (31), one can include the relaxation effect. In that case, Eq. (30) becomes GVV (E) 5
r˜ ( ´˜ ) E ]] d´˜ E 2 ´˜ V2
(32)
Here, ´˜ is the sum of two valence-hole energies. Including in Eq. (32) the screened hole–hole interaction, U˜ VV , given by U˜ VV (E) 5 UVV 2 2 Re SV (E)
(33)
to infinite order by the ladder approximation (UVV is the bare hole–hole interaction), one obtains the two
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
valence-hole spectral function (the density of two valence-hole states)
Im S˜ V (E) 5
1 D A V 2 (E) 5 ] ]]]]]]] p (1 1 U˜ F )2 1 (U˜ D)2 VV
Here F(E) and D(E) are the real part and the imaginary part of Eq. (32), respectively. When the potential binding condition 1 1 U˜ VV F 5 0 and D 5 0
(35)
is satisfied, the two-hole state becomes a bound state. Here D 5 0 means r˜ V 2 5 0. With an increase of U˜ VV , the two-hole bound state dominates (see also Refs. [13,14]). When U˜ VV becomes comparable with the bandwidth of r˜ V 2 , Eq. (34) becomes the imaginary part of Eq. (3) divided by p. We now consider the localization of three valencehole state created by final state shakeoff upon Auger decay. Assuming that the dominant part of the valence hole self-energy in the presence of an extra valence hole is due to the shakeoff process, one obtains the self-energy given by
SV (E) 5
r˜ ( ´˜ ) E uV u r (´ ) FE ]]] d´˜ G d´ E 1 ´ 2 ´˜ V3
2
V
s
s
I E uV u r (´ ) ]]]]]]]] d´ ˜ (1 1 3U R) 1 (3U˜ I) 2
V
s
(36)
Here, rs is the density of the unoccupied states for a shakeoff electron. ´s is the shakeoff electron energy. E is the two-hole energy. VV is the energy-independent valence shakeoff matrix element. r˜ V 3 is the convolution of the effective density of the valencehole states given by
E r˜
V2
( ´˜ 2 ´ 9) r˜ V (´ 9) d´ 9
(37)
´˜ is now the sum of three valence-hole energies. When U˜ VV is included in the valence-hole selfenergy given by Eq. (36), to infinite order by the ladder approximation, the real part and imaginary part of the renormalized self-energy S˜ V (E) becomes Re S˜ V (E) 5
E
s
2
VV
R 1 3U˜ VV (R 2 1 I 2 ) 2 uVV u rs (´s ) ]]]]]]]] d´s (1 1 3U˜ VV R)2 1 (3U˜ VV I)2
(38)
2
s
(39)
VV
Here, R and I are the real part and the imaginary part of the function inside the parentheses in Eq. (36), respectively. In Eq. (39), when 1 1 3U˜ VV R 5 0 and I 5 0 are satisfied, the three-hole state becomes a bound state. With an increase of U˜ VV , the bound state dominates more than the band state. When U˜ VV becomes comparable with the bandwidth of r˜ V 3 , Eqs. (38) and (39) become the real part and the imaginary part of the self-energy given by Eq. (12), respectively. When Eq. (35) and 1 1 3U˜ VV R 5 0 (I50) are satisfied, both the two-hole state and the three-hole state become bound states. So far, we considered the localization of the three-hole state created by final state shakeoff upon Auger decay. When the three holes are created by the Auger decay of the initial core shakeoff state, we consider the core-hole selfenergy in the presence of an extra (valence) hole, the imaginary part of which is given by Eq. (28). Then we consider the three valence-hole Green’s function, the imaginary part of which divided by p provides the density of three-hole states A lmv in Eq. (28). The Green’s function is given by
s
r˜ V 3 ( ´˜ ) 5
s
(34)
VV
35
GV 3 (E) 5
r˜ ( ´˜ ) E ]] d´˜ , E 2 ´˜ V3
(40)
Here E is the three-hole energy. Defining the real part and the imaginary part of Eq. (40) by R and I, one obtains the potential binding condition 1 1 3U˜ VV R 5 0 and I50 for the localization of the threehole state. For the sake of simplicity, the particlehole interaction, which can be included in the same manner as the hole–hole interaction, is neglected. The inclusion of the particle-hole interaction leads to the singularity that provides a final shakeup bound state.
5. Conclusion The Auger satellite spectral line profile depends on both initial state and final state. The spectral line profile of the three-hole satellite state created by the final state shakeoff upon the Auger electron emission
36
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
of the predominantly single core-hole main-line state differs from that of the three-hole state due to the Auger decay of the initial shakeoff satellite state. The former is essentially the convolution of the main line state part of the initial core-hole spectral function with the shakeoff satellite part of the two-hole spectral function, weighted by the unrenormalized partial decay rate ratio. The latter is more complicated. The potential binding conditions for the localization of the two- and three-hole states when self-energy correction, such as the shakeoff excitations and the screening of the bare hole–hole interaction by the particle-hole pair excitations, are taken into account, are derived.
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On the satellite in the Auger electron spectrum Masahide Ohno* Advanced Physics Research Laboratory, 2 -8 -5 Tokiwadai, Itabashi-ku, Tokyo, 174 -0071, Japan Received 4 March 1999; received in revised form 11 May 1999; accepted 11 May 1999
Abstract The satellite line profile in the C–VV Auger electron spectroscopy spectrum depends upon how the initial core hole state and the final multiple hole state are created. Even if the numbers of the holes in the satellite states are the same, the Auger electron spectroscopy spectral profiles of the final multiple hole states are different. The localization of the three hole satellite state is discussed. 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Photoionization is often a very rapid ‘instantaneous’ application of the perturbation (i.e. in a short time compared with the periods of transition from a given state to other state). In contrast to the adiabatic limit where the entropy of the system remains unchanged, i.e. the process is reversible, the sudden limit is the nonadiabatic limit where the process is irreversible. The nonadiabatic ionization process induces a number of correlated ionized states, because of a conservation of oscillator strength, which requires that any suppression of the direct transition be compensated for by the matrix element to other excited states of the system, in which additional electrons are excited from their unperturbed states. Friedel [1] called such additional excitations the shakeoff. The shakeup / off satellites in the core (valence) electron ionization spectra have been studied to elucidate the ground state electronic structures. However, the satellite observed in the Auger electron *Corresponding author.
spectroscopy (AES) spectrum has not been of much concern, except in the case of C–VV AES spectra of the elements around Cu (free atoms, metals, compounds) where the final three valence-hole state can be created by different decay channels. These AES spectra have been the subject of intensive study [2–8]. If the initial core shakeoff satellite state does not relax to the initial core main-line state on the time-scale of core-hole decay, both states can decay independently. Then, the initial core shakeoff satellite state can decay to the three valence-hole state (CV–VVV), while the final state valence shakeoff upon Auger decay of the initial main-line state results in the three valence-hole state too (C–VVV). Coster–Kronig (CK) decay preceded Auger decay results also in the three valence-hole state (C–CV– VVV). Recently, Auger photoelectron coincidence spectroscopy (APECS) has begun to show its potential for studies of electron correlations [8–11]. Using APECS, one can determine the correlations between the initial state ionizations and the Auger final states by measuring the photoelectron (or Auger electron) in coincidence with the specified Auger final state (or
0368-2048 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 99 )00032-8
30
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
specified initial core-hole ionized state). A recent APECS study of C–VV Auger transition of the 3d transition metals and related systems showed the importance of the decay of the initial shakeup / off satellite state to the three hole-state [9,11]. Thurgate [9] stated that it should be recognized that there is no reason that the initial state and final state shakeup / off should produce the same effect on the Auger line shape; the initial state shakeoff produces a spectator hole that may influence the valence band structure and interact with the out-going Auger electron, while the final state shakeoff produces an extra hole during the Auger electron emission process. The final state may be the same, but the initial state is different. The AES spectral profile is essentially described by the convolution of the initial core-hole spectral function and the final state spectral function (density of the final states), weighted by the partial decay rate ratio [12]. Thus, one should recognize that the AES line profile must be different when the initial state and the final state are created by different processes. In this paper, it will be theoretically shown that the Auger satellite line profile depends on how it is created. The localization of the three-hole satellite state will be discussed. We take into account the screening of the bare hole hole interaction and the relaxation (fluctuations) and decay of hole(s) by renormalizing the multiple-hole Green’s function.
one-particle shakeoff state. For the sake of simplicity, we neglect the multiplet structure of the final state. The valence-hole spectral function is given by Im SV (E) 1 A V (E) 5 ] ]]]]]]]]]] p (E 2 ´V 2 Re SV (E))2 1 (Im S V (E))2 (2) We consider the two-hole Green’s function by renormalizing each hole’s propagator by the selfenergy in the presence of an extra hole and screening the bare hole-hole interaction by the hole-particle pair excitations. Then, the two-hole Green’s function for the holes i and j (i ±j) becomes Gij (E) 5sE 2 ´i 2 ´j 1 Uij 2 Si (E) 2 Sj (E) 2 Sij (E) 2 Sji (E)d 21
(3)
Here
Si (E) 2
5S
l,m
uV (´)u E ]]]]]]]]]]] d´ E 2´ 2´ 2´ 1U 1U 1U 1´ iml ´
l
m
j
lm
mj
lj
(4)
Sj (E) 5S
l,m
E
uVjml ´ (´)u 2 ]]]]]]]]]]] d´ E 2 ´l 2 ´m 2 ´i 1 Ulm 1 Umi 1 Uli 1 ´ (5)
2. AES spectral profiles We consider the case where the dominant part of the satellite is due to the shakeoff process and the valence hole(s) is localized. In Section 4, we extend the formalism to the case where the hole(s) is created in the valence band. For the valence electron ionization, the valence-hole self-energy is given by
Sij (E) 5S l
1S l
uV u r (´) E ]]]]]] d´ E 1 ´ 2 2´ 1 U V
s
V
(1)
VV
Here ´V is the Koopmans valence hole energy, ´ is the shakeoff electron energy, rs is the density of the unoccupied states for the shakeoff electron. UVV is bare valence hole-hole interaction and Vv is the energy-independent coupling matrix element between the single valence-hole state and the two hole
uV (´)V (´)u E ]]]]]]]]]] d´ E 1´ 2´ 2´ 2´ 1U 1U 1U ili ´
i
Sji (E) 5S l
2
SV (E) 5
1S l
E
uVilj ´ (´)u 2 ]]]]]]]] d´ E 1 ´ 2 ´l 2 2´j 1 2Ulj 1 Ujj
E
l
jlj ´
j
il
lj
(6)
ij
uVjli ´ (´)u 2 ]]]]]]]] d´ E 1 ´ 2 ´l 2 2´i 1 2Uli 1 Uil
uV (´)V (´)u E ]]]]]]]]]] d´ E 1´ 2´ 2´ 2´ 1U 1U 1U jlj ´
i
l
ili ´
j
il
lj
(7)
ij
Here the ‘large S’ denotes the summation over the hole indices. Eq. (4) describes the relaxation and decay of hole i( j) in the presence of an extra hole j(i). Eqs. (6) and (7) provide the screening of the bare hole-hole interaction between i and j by the hole particle pair excitations. V is the Coloumb
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
matrix element and includes the exchange term implicitly. U is the bare hole-hole interaction, included to infinite order by the ladder approximation. Eqs. (4)2(7) include the relaxation (decay) of the holes i and j or the screening of the bare hole-hole interaction between i and j to second order. The bare hole-hole interaction among the final state holes is included to infinite order, but the relaxation (decay) of the holes and the screening of the bare hole-hole interaction are neglected. One can include the relaxation of the final state holes and the screening of the bare hole-hole interaction by introducing the threehole (one particle) propagator and repeat the same renormalization procedure. However, as a simpler approximation for the relaxation and decay of the final state holes, one may replace the Koopmans hole energies in the denominators of Eqs. (4)–(7) by the renormalized hole energies. El 5 ´l 1 Dl (El ) 1 iGl (El )
(8)
31
A V 2 (E) 5 2 Im SV (E) 1 ] ]]]]]]]]]]]]]] p (E 2 2´V 1 UVV 2 4 Re SV (E))2 1 (2 Im SV (E))2 (11) Here SV (E) is the valence-hole (shakeoff) selfenergy given by 2
SV (E) 5
uV u r (´) E ]]]]]]]] d´ E 1 ´ 2 3E 1 3U 2 6D V
s
V
VV
(12)
V
In the same manner, one can obtain the three-hole spectral function (the density of three-hole states). A V 3 (E) 5 3 Im SV (E) 1 ] ]]]]]]]]]]]]]] p (E 2 3´V 1 3UVV 2 9 Re SV (E))2 1 (3 Im SV (E))2 (13)
Here D and G are the real and imaginary part of the hole self-energy given by
Sl (E) 5n,m S
E
uVlnm ´ (´)u 2 ]]]]]]] d´ E 1 ´ 2 ´n 2 ´m 1 Unm
(9)
El is the solution of the Dyson equation given by El 5 ´l 1 Re Sl (El )
(10)
For the screening of the bare hole-hole interaction in the denominator of Eqs. (4)–(7), one can replace the bare hole-hole interaction, Uij , approximately by Uij 2 Di 2 Dj . Here, D is given by Eq. (8). We neglect the effect of the presence of an extra hole. When i 5 j, the first term and the second term of Eq. (6) become identical. When i 5 j, the imaginary part of the self-energy given by Eq. (6) becomes identical to that given by Eq. (4) from a viewpoint of unitary expansion. Thus, when i5j, the imaginary part of the self-energy by Eqs. (6) and (7) does not contribute. The decay width of the two-hole is approximately given by twice the single-hole decay width. When the two holes are created in the same valence shell (v), the two-hole spectral function (density of twohole states) becomes
So far, the interaction between the particle (shakeup / off electron) and the holes in the final state has been neglected. The interaction, U, can be included to infinite order by the ladder and ring approximation. By defining the real and imaginary part of the selfenergy given by Eq. (12) as follows
SV (E) 5 F 1 iD
(14)
The real and imaginary part of self-energy renormalized by including U to infinite order by the ladder and ring approximation become F 2 U(F 2 1 D 2 ) Re SV (E) 5 ]]]]]] (1 2 UF )2 1 (UD)2
(15)
D Im SV (E) 5 ]]]]]] (1 2 UF )2 1 (UD)2
(16)
We now consider the shakeoff upon the core electron ionization. The unrenormalized core-hole self-energy for the shakeoff is given by 2
S sc (E) 5
uV u r (´ ) E ]]]]]]] d´ E 1´ 2´ 2´ 1U c
s
s
C
s
V
CV
s
(17)
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
32
Here ´C , ´V , UCV and ´s are the Koopmans core-hole energy, the Koopmans valence-hole energy, the bare core hole–valence hole interaction and the shakeoff electron energy, respectively. rs (´s ) is the density of the unoccupied states for the shakeoff electron and Vc is the energy-independent coupling matrix element between the single core-hole state and the two-hole one-particle shakeoff state. When we take into account the decay and relaxation of the two-hole and the screening of the bare hole-hole interaction, Eq. (17) becomes
S sc (E) 5
E uV u r (´ )G 2
I mAES (´A )v 5
F
E A (´ 2 v) m
p uVCVV´A (´A )u 2 ]]]] A 2 (´ 1 ´ 2 v ) Im S Ac (´ 2 v ) V A
G
d´
(22)
Here, ´ and v are the photoelectron energy and the incident photon energy, respectively. A m is the mainline state part of the initial core-hole spectral function given by 1 A m (´ 2 v ) 5 ] p
(18)
Im S Ac (´ 2 v ) ]]]]]]]]]]]]] A (´ 2 v 2 ´c 2 Re S c (´ 2 v ) 2 Re S sc (´ 2 v ))2 1 (Im S Ac (´ 2 v ))2
Here Gcv is the one core-hole one valence-hole Green’s function (Eq. (3) for i 5 c and j 5 v). We consider also the Auger decay c 21 2 l 21 m 21 ´A of the initial core-hole (c). The corresponding unrenormalized self-energy of the core-hole is given by
(23)
C
s
s
CV
(E 1 ´s ) d´s
uVclm ´A (´A )u 2 S (E) 5 S ]]]]]]] d´A l,m E 1 ´A 2 ´l 2 ´m 1 Ulm A c
(19)
l,m
E uV
clm ´A
(´A )u 2 Glm (E 1 ´A ) d´A
(20)
Here Glm is the two-hole Green’s function (i 5 l, j 5 m in Eq. (3)). The core-hole Green’s function is given by Gc (E) 5 [E 2 ´c 2 S Ac (E) 2 S sc (E)] 21
Im S cA (´ 2 v ) 5 S
l,m
Here Vclm ´A is the energy-dependent Auger decay matrix element. When we take into account the decay and relaxation of two holes, l and m, and the screening of the bare hole-hole interaction, Eq. (19) becomes
S Ac (E) 5 S
The line profile of A m is broadened by the Auger decay of the core-hole. A v 2 is the final two valencehole spectral function (Eq. (11)). The imaginary part of the core-hole’s self-energy is given by (see Eq. (20))
(21)
We consider the case where two initial core-hole states (main line and shakeoff satellite) can decay independently. Moreover, the couplings to the same final state from different initial core-hole states are so small that the interference terms are negligible. The AES spectral line profile for the decay of the main-line state is given by
E p uV
clm ´A
2 v ) d´A
(´A )u 2 A lm (´A 1 ´ (24)
The AES spectral profile is given by the convolution of the initial state spectral function and the final state spectral function, weighted by the unrenormalized partial decay rate ratio. Here, the unrenormalized means the neglect of the final two hole state lifetime and interaction. When the AES spectrum is integrated with the Auger electron energy, the factor inside the parentheses becomes the renormalized partial decay rate ratio. The renormalized partial decay rate is the convolution of the unrenormalized partial decay rate with the two-hole spectral function. As the width of the two-hole spectral function is negligible compared to the ‘width’ of the density of the unoccupied states for the Auger electron, the effect is small. However, the CK decay electron energy is much smaller compared to the Auger electron energy. Moreover, the ‘width’ of the unoccupied states for the CK electron is much smaller than that for the Auger electron. Thus, the broadening may affect the decay rate near the double ionization threshold. When all decay channels are
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
considered, the total integrated ‘decay’ spectral intensity becomes the initial main line state spectral intensity. Here, ‘decay’ means both radiationless and radiation decays. For the Auger decay of the initial shakeoff satellite state, the AES spectrum becomes AES
Is
(´A )v 5
A (´ 2 v ) E ]]]] Im S (´ 2 v ) SE p uV u r (´ )A s
s c
2
c
F
s
s
cv
(´s 1 ´ 2 v )
˜ ´ )u 2 A 3 (´ 1 ´ 1 ´ 2 v ) p uV( A V s A ]]]]]]]]]]] ˜ Im Sc (´s 1 ´ 2 v ) 1 Im S˜ V (´s 1 ´ 2 v )
G D
d´s d´ (25)
Here A s is the satellite part of the initial core-hole spectral function given by 1 A s (´ 2 v ) 5 ] p Im S sc (´ 2 v ) ]]]]]]]]]]]]] A (´ 2 v 2 ´c 2 Re S c (´ 2 v ) 2 Re S sc (´ 2 v ))2 1 (Im S 2c (´ 2 v ))2
Im S˜ c (´s 1 ´ 2 v ) 5 S
l,m
E puV˜
clm ´A
33
(´A )u 2 A lmv (´s
1 ´A 1 ´ 2 v ) d´A
(28)
V˜ is the Auger decay matrix element for the core hole in the presence of a spectator valence hole. Thus, the self-energy S˜ c differs from the self-energy introduced by Eq. (19). The factor inside the parentheses becomes the renormalized partial decay rate ratio when the AES spectrum is integrated with the Auger electron energy. When all decay channels, including the decay of the valence hole, are considered, the factor inside the bracket becomes the imaginary part of the shakeoff self-energy (Eq. (27)). Then the total integrated decay spectral intensity becomes the integrated initial shakeoff satellite intensity. Compared to the AES spectral profile for the decay of the initial main-line state, the one from the initial shakeoff satellite state is much more complicated. So far, in the core hole self-energy, the particle-hole(s) interaction is neglected for the sake of simplicity. However, it can be included, as in Eqs. 15 and 16.
(26) The imaginary part of the shakeoff self-energy S sc (´ 2 v ) is given by (Eq. (18)) Im S sc (´ 2 v ) 5
E p uV u r (´ )A 2
C
s
s
CV
(´s 1 ´ 2 v ) d´s
3. Three hole state in the AES spectrum To reach the final three-hole state, there are two possible channels.
(27) A CV is the one core hole–one valence hole pair spectral function given by the imaginary part of GCV divided by p (i 5 c and j 5 v in Eq. (3)). V˜ is the C–VV Auger decay matrix element in the presence of a spectator valence hole. Note that the imaginary part of the shakeoff self-energy is given by the convolution of the density of the unoccupied states for the shakeoff electron and the core hole–valence hole pair spectral function A CV . Thus, the density of the unoccupied states is broadened by the lifetime of the core hole–valence hole pair. In general, the width of the former is much larger than the latter. Thus, the broadening effect is negligible. The imaginary part of the core hole self-energy S˜ c is given by
Channnel A: The final state shakeoff upon the Auger decay of the initial main line state, which is predominantly single core-hole state. Channel B: The Auger decay of the initial shakeoff satellite state, which is predominantly one core-hole and one valence hole state. For channel A, the AES spectrum is given by the satellite part of the AES spectrum given by Eq. (22). For channel B, the AES spectrum is given by the main line part of the AES spectrum given by Eq. (25). Thus, the AES spectral profiles for channels A and B are different not only in the Auger energies but also in the spectral profiles.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
34
For L 2 –L 3 V–VVVCK, decay preceded Auger decay, Eq. (25) becomes I AES L 2 – L 3 V–VVV (´A )v 5
SE
F
EA
L2
1 (´ 2 v ) ]]]] Im SL 2 (´ 2 v )
puVCK ( ´˜ )u 2 A L 3 V ( ´˜ 1 ´ 2 v )
puVA (´A )u 2 A v 3 ( ´˜ 1 ´A 1 ´ 2 v ) ]]]]]]]]]]] Im S˜ L 3 ( ´˜ 1 ´ 2 v ) 1 Im S˜ V ( ´˜ 1 ´ 2 v )
GD
d´˜ d´ (29)
A L 2 is the main line state part of the L 2 core-hole spectral function. Im SL 2 is the imaginary part of the L 2 core-hole self-energy. A L 3 V is the L 3 V two-hole spectral function. VCK is the energy-dependent L 2 – L 3 VCK decay matrix element and ´˜ is the corresponding CK decay electron energy. VA is the L 3 V– VVV Auger decay matrix element. S˜ L 3 is the selfenergy for the L 3 core-hole in the presence of a valence hole. If we compare Eq. (29) with Eq. (25) for the case of L 3 V (shakeoff)–VVV Auger decay, the following factors are different; the initial corehole spectral function, the core-hole self-energy and the shakeoff matrix element. Then the AES spectral line profile for the three-hole states by L 2 –L 3 V– VVV decay is different from that by the L 3 V–VVV decay. The APECS spectrum measured in coincidence with a particular initial core-hole state shows the spectral function of the final state reached by the Auger decay of the selected initial state, while the APECS spectrum measured in coincidence with a particular Auger final state shows the spectral function of the initial state for the Auger decay (the spectral line width will become narrower when the final state spectral width is narrower than the initial state one). The APECS can be used to separate the initial and final state events. However, one cannot obtain separately the AES spectral profiles for channels A and B. For channel A, the APECS spectrum essentially shows the final two-hole spectral function. If the final satellite state is well separated from the final main line state, one can obtain the main line part and the satellite part of the final state two-hole spectral function. For channel B, the situation is different.
The APECS spectrum shows essentially the final three-hole spectral function convoluted by the one core-hole one valence-hole shakeoff state spectral function. If the presence of multiplet levels in the initial shakeoff state is substantial, the APECS spectrum can be quite complicated. In that case, it is necessary to scan each initial shakeoff multiplet state so that one can decompose the APECS spectrum by the initial state events.
4. Localization of the multiple hole state So far, the valence holes are assumed to be localized. We consider the cases when holes are created in a valence band. The unperturbed two valence-hole Green’s function (‘unperturbed’ means before the valence hole-hole interaction is switched on) is given by GVV (E) 5
r (´) E ]]]]] d´ E 2 ´ 2 2S (E) V2
(30)
V
Here, rV 2 (´) is the self-convolution of the Koopmans density of the valence hole states given by
rV 2 (´) 5
E r (´ 2 ´9)r (´9) d´9 V
V
(31)
´ is the sum of two Koopmans valence-hole energies. The relaxation and decay of two holes could be included by the hole self-energy SV (E). In Eq. (30), the relaxation effect on the density of the hole states is neglected by using the Koopmans density of the states (Eq. (31)). By using the effective density of the hole states, r˜ V , which corresponds to the valence hole spectral function (main line state), instead of rV in Eq. (31), one can include the relaxation effect. In that case, Eq. (30) becomes GVV (E) 5
r˜ ( ´˜ ) E ]] d´˜ E 2 ´˜ V2
(32)
Here, ´˜ is the sum of two valence-hole energies. Including in Eq. (32) the screened hole–hole interaction, U˜ VV , given by U˜ VV (E) 5 UVV 2 2 Re SV (E)
(33)
to infinite order by the ladder approximation (UVV is the bare hole–hole interaction), one obtains the two
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
valence-hole spectral function (the density of two valence-hole states)
Im S˜ V (E) 5
1 D A V 2 (E) 5 ] ]]]]]]] p (1 1 U˜ F )2 1 (U˜ D)2 VV
Here F(E) and D(E) are the real part and the imaginary part of Eq. (32), respectively. When the potential binding condition 1 1 U˜ VV F 5 0 and D 5 0
(35)
is satisfied, the two-hole state becomes a bound state. Here D 5 0 means r˜ V 2 5 0. With an increase of U˜ VV , the two-hole bound state dominates (see also Refs. [13,14]). When U˜ VV becomes comparable with the bandwidth of r˜ V 2 , Eq. (34) becomes the imaginary part of Eq. (3) divided by p. We now consider the localization of three valencehole state created by final state shakeoff upon Auger decay. Assuming that the dominant part of the valence hole self-energy in the presence of an extra valence hole is due to the shakeoff process, one obtains the self-energy given by
SV (E) 5
r˜ ( ´˜ ) E uV u r (´ ) FE ]]] d´˜ G d´ E 1 ´ 2 ´˜ V3
2
V
s
s
I E uV u r (´ ) ]]]]]]]] d´ ˜ (1 1 3U R) 1 (3U˜ I) 2
V
s
(36)
Here, rs is the density of the unoccupied states for a shakeoff electron. ´s is the shakeoff electron energy. E is the two-hole energy. VV is the energy-independent valence shakeoff matrix element. r˜ V 3 is the convolution of the effective density of the valencehole states given by
E r˜
V2
( ´˜ 2 ´ 9) r˜ V (´ 9) d´ 9
(37)
´˜ is now the sum of three valence-hole energies. When U˜ VV is included in the valence-hole selfenergy given by Eq. (36), to infinite order by the ladder approximation, the real part and imaginary part of the renormalized self-energy S˜ V (E) becomes Re S˜ V (E) 5
E
s
2
VV
R 1 3U˜ VV (R 2 1 I 2 ) 2 uVV u rs (´s ) ]]]]]]]] d´s (1 1 3U˜ VV R)2 1 (3U˜ VV I)2
(38)
2
s
(39)
VV
Here, R and I are the real part and the imaginary part of the function inside the parentheses in Eq. (36), respectively. In Eq. (39), when 1 1 3U˜ VV R 5 0 and I 5 0 are satisfied, the three-hole state becomes a bound state. With an increase of U˜ VV , the bound state dominates more than the band state. When U˜ VV becomes comparable with the bandwidth of r˜ V 3 , Eqs. (38) and (39) become the real part and the imaginary part of the self-energy given by Eq. (12), respectively. When Eq. (35) and 1 1 3U˜ VV R 5 0 (I50) are satisfied, both the two-hole state and the three-hole state become bound states. So far, we considered the localization of the three-hole state created by final state shakeoff upon Auger decay. When the three holes are created by the Auger decay of the initial core shakeoff state, we consider the core-hole selfenergy in the presence of an extra (valence) hole, the imaginary part of which is given by Eq. (28). Then we consider the three valence-hole Green’s function, the imaginary part of which divided by p provides the density of three-hole states A lmv in Eq. (28). The Green’s function is given by
s
r˜ V 3 ( ´˜ ) 5
s
(34)
VV
35
GV 3 (E) 5
r˜ ( ´˜ ) E ]] d´˜ , E 2 ´˜ V3
(40)
Here E is the three-hole energy. Defining the real part and the imaginary part of Eq. (40) by R and I, one obtains the potential binding condition 1 1 3U˜ VV R 5 0 and I50 for the localization of the threehole state. For the sake of simplicity, the particlehole interaction, which can be included in the same manner as the hole–hole interaction, is neglected. The inclusion of the particle-hole interaction leads to the singularity that provides a final shakeup bound state.
5. Conclusion The Auger satellite spectral line profile depends on both initial state and final state. The spectral line profile of the three-hole satellite state created by the final state shakeoff upon the Auger electron emission
36
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 29 – 36
of the predominantly single core-hole main-line state differs from that of the three-hole state due to the Auger decay of the initial shakeoff satellite state. The former is essentially the convolution of the main line state part of the initial core-hole spectral function with the shakeoff satellite part of the two-hole spectral function, weighted by the unrenormalized partial decay rate ratio. The latter is more complicated. The potential binding conditions for the localization of the two- and three-hole states when self-energy correction, such as the shakeoff excitations and the screening of the bare hole–hole interaction by the particle-hole pair excitations, are taken into account, are derived.
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