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Many-electron effect in the Si K-LL resonant Auger-electron spectroscopy spectra of the Si delta layer in GaAs
Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
Many-electron effect in the Si K-LL resonant Auger-electron spectroscopy spectra of the Si delta layer in GaAs Masahide Ohno Quantum Science Research, 2-8-5 Tokiwadai, Itabashi-ku, Tokyo 174-0071, Japan Received 6 December 2005; received in revised form 6 April 2006; accepted 10 April 2006 Available online 5 July 2006
Abstract The Si K-LL resonant Auger-electron spectroscopy (RAES) spectra of silicon delta dopped layers in GaAs with very thin capping layers show both normal Auger decay and resonant Auger decay, when the core-level electron is excited to the conduction band. The resonant Auger peak kinetic energy (KE) shows no dispersion with photon energy, except when excited by the highest energy photons [M.D. Jackson, J.M.C. Thornton, D. Lewis, A. Robinson, M. Fahy, A. Aviary, P. Weightman, Phys. Rev. B71 (2005) 075313]. The RAES spectra are analyzed using a many-body theory. The presence of resonant Auger decay and no dispersion of resonant Auger peak KE with photon energy is explained in terms of the relaxation of a metastable excited core-hole state to a stable one on the time scale of core-hole decay. The excited electron in the conduction band either delocalizes rapidly leaving the ionized Si to decay by a normal Auger decay or drops to a state localized in the Si delta layer before the core-hole decays so that the RAES spectrum has both normal Auger decay and resonant Auger decay. As a result of the relaxation, the resonant Auger peak KE does not show any dispersion with photon energy. The variations with photon energy of the normal or resonant Auger peak intensity, KE, and width are explained in a consistent manner by a many-body theory. © 2006 Elsevier B.V. All rights reserved. Keywords: Si delta dopped layers in GaAs; Resonant Auger-electron spectroscopy (RAES); Normal Auger decay; Resonant Auger decay; Resonant radiationless Raman scattering; Core-hole lifetime; Core-hole relaxation
1. Introduction As a result of energy conservation of a coherent one-step process of resonant excitation, autoionization decay channels show two distinct characteristic resonant Auger–Raman effects, i.e., linear dispersion of the kinetic energy (KE) of the decay channels with excitation energy and line narrowing [1–6]. When the excited electron is present during the core-hole decay, the decay spectra show the resonant Auger–Raman effects, while those where the excited electron delocalizes before the decay will show normal Auger decay [7–11]. When the core-level electron is excited to unoccupied states the bandwidth of which is much smaller than the core-hole lifetime width, above the absorption edge the excess energy of the photon will be transferred completely to the Auger electron so that the resonant Auger-electron spectroscopy (RAES) line appears at the constant binding energy. However, when the core-level electron is excited to unoccupied states the bandwidth of which is much larger than the core-hole lifetime width, above the absorption edge the excess energy of the photon will be absorbed by the 0368-2048/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2006.04.001
system probed so that the RAES line appears at constant KE, however, the line shape is still modulated by the partial density of unoccupied states. Even when the core-level electron is excited into unoccupied delocalized states, if there is a mechanism by which the excited electron can transit to an unoccupied localized state lying below the unoccupied band states on the time scale of core-hole decay, the resonant Auger decay will be retrived even above the absorption edge, in addition to the normal Auger decay. Recently Jackson et al. [12] measured the K X-ray absorption spectroscopy (XAS) spectra and the RAES spectra of silicon delta dopped layers in GaAs with very thin capping layers. In the present paper we focus on the RAES spectra of the 0.01 ML sample. The Si K-edge XAS spectrum shows two distinct features: a pre-edge to the absorption which occurs at an excitation energy of 1838.8 eV, and a second peak at about 1841 eV (Fig. 2(a) in Ref. [12]). The former XAS peak arises from a localized Si state at the Si layer and does not involve the electronic structure of the GaAs. The second peak arises from the maximum intensity of the conduction band of GaAs which occurs
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
at 1.4 eV above the bottom of the conduction band (=1839.7 eV [12]). The energy of the second peak corresponds to the Si 1s ionization energy. When excited far above threshold the Si KL23 L23 (=KLL) AES spectra show the characteristic multiplet structure which consists of a strong component arising from the 1 D term of the two 2p hole final state which was observed at 2 a KE of 1606 eV and a weak component which arises from the 1 S term and occurs 6 eV lower in KE. When excited by photons close to threshold the Si KLL RAES spectra do not show this characteristic multiplet structure and vary markedly with photon energy (Fig. 3(a) in Ref. [12]). The spectrum consists of two main peaks and of these the one to lower KE evolves into the D term observed at a constant KE of 1606 eV when excited by photons far above threshold. They associate the peak to higher KE with a resonant Auger decay as it is observed when excited by radiation close to the K threshold and it exhibits narrower width than the D term. The maximum intensity of this feature is observed when the spectrum is excited by photons corresponding to the maximum of the first peak in the XAS spectrum, i.e., the Si 1s electron is excited into a localized state associated with the delta layer (Fig. 4(a) in Ref. [12]). The resonant Auger decay requires a long-lived localized excited state, i.e., the characteristic of a state localized at a delta layer in the band gap of a semiconductor [12]. The resonant Auger peak remains at a constant KE (1611 eV, the same as the one measured at the first XAS line peak (1838.8 eV)) over the 2 eV photon excitation range (1839–1841 eV) for which it is intense (Figs. 3(a) and 6 in Ref. [12]). The resonant Auger peak shows no dispersion with photon energy except when excited by the highest energy photons. This is in contrast to most other resonant Auger decays, which show a positive dispersion with photon energy ([3–6] and references in [12]). Jackson et al. [12] suggested that a likely explanation for this behavior is that in the delta layer the unoccupied state is very narrow and shows no dispersion in energy. With these characteristics it can only be excited by a narrow range of photon energies, a range that may be narrower than the resolution of their measurement. They pointed out that at higher photon
energies it is possible that the localized state can be occupied by electrons, which are first excited into the GaAs conduction band and which then transfer their energy to the resonant Auger electron when dropping into the localized state. While this would provide a mechanism to increase the resonant Auger-electron KE, as exhibited in the spectrum excited by the highest energy photons (Fig. 6 in Ref. [12]), they argued that there will be little mixing between the localized state in the Si delta layer and the GaAs conduction band, and the transition will be weak [12]. The electron excited to the bottom of the GaAs conduction band would be expected to delocalize from the excitation site leaving the ionized Si to decay by a normal Auger decay [12]. The normal Auger peak KE shows a negative dispersion with photon energy (Fig. 6 in Ref. [12]), which could be due to the postcollision interaction (PCI) effect [12]. The resonant Auger peak width shows an increase with photon energy, whereas the normal Auger peak width shows an opposite trend. Both widths merge at the second XAS peak energy (Fig. 5 in Ref. [12]). The variation with photon energy of the resonant or normal Auger peak intensity is shown in Fig. 4 in Ref. [12] The variations with photon energy of the resonant or normal Auger peak KE, width and intensity require an explanation. In the present paper by a many-body theory we shall discuss the RAES spectral behavior in terms of the transition of an excited electron in delocalized states to a localized state on time scale of core-hole decay. 2. Theory 2.1. Resonant excitation to a localized state and resonant Auger decay We consider the resonant core-level electron excitation to an unoccupied localized state and subsequent resonant Auger decay. The RAES spectrum is (Fig. 1): d2 σ = |P˜ ca |2 |Gca (ω)Vsf (εA )|2 Af (εA − ω) dω dεA
(1)
f
Fig. 1. The resonant Auger decay: (a) The radiationless participant decay of the resonantly excited core-hole state in which the excited electron occupies an unoccupied localized state such as the one in the Si delta layer. The core-hole propagator of the excited core-hole state is dressed by the core-hole self-energy by both radiationless and radiation decay. (b) The radiationless spectator decay of the resonantly excited core-hole state defined in the same manner as in (a). The excited electron remains localized as a spectator during the core-hole decay. When the screening of the two-hole potential in the final state by the spectator electron is effective, and the resonant Auger-electron KE is small, the presence of the spectator electron affects the resonant Auger decay rate so that the term “spectator” is not appropriate anymore. This is not the case in the present case.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
P˜ ca is the dipole-transition matrix element of the core-level electron excitation to an unoccupied localized state; ω the incident photon energy; Gca the core-hole excited-electron pair (or the f excited core-hole state) propagator; Vs the radiationless participant or spectator decay matrix element of the excited core-hole state; εA the KE of the electron emitted by the core-hole decay and Af is the density (or the spectral function) of the final states broadened by their hole lifetime. The final state by the radiationless participant decay is a single-hole (1h) state, while that by the radiationless spectator decay is a two-hole one-particle (2h1p) state. The photon energy distribution and the spectral function of the electron analyzer can be additionally included in Eq. (1). The core-hole excited-electron pair propagator Gca is Gca (ω) = (εa − εc − ω − iΓs )−1
(2)
Here εa is the energy of an unoccupied localized state in the presence of the core-hole; εc the core-hole energy and Γ s is the lifetime width of the excited core-hole state by the radiation and radiationless decays. Eq. (1) becomes f
d2 σ π|Vs (εA )|2 = Io (ω) Af (εA − ω) dω dεA Γs
(3)
f
Io (ω) is the XAS spectrum for the resonant core-level electron excitation to an unoccupied localized state. Io (ω) = |P˜ ca |2 Aca (ω)
(4)
Aca is the spectral function of the resonantly excited core-hole state, i.e., the imaginary part of the core-hole excited-electron pair (or the excited core-hole state) propagator (Eq. (2)) divided by π. The RAES (resonant Auger decay) spectrum is a superposition of the product of the XAS spectral intensity at a selected photon energy, the branching ratio of the unrenormalized radiationless (spectator or participant) decay rate of the excited core-hole state, and the density (the spectral function) of the 1h or 2h1p final states. The RAES peak appears at the constant binding energy. The RAES peak KE shows a positive dispersion with photon energy. The RAES peak width is as narrow as the final-state width. The peak width narrowing is analogous to that of the normal Auger peak, when the Auger electrons are collected in coincidence with photoelectrons of a fixed KE. The coincidence AES spectrum is a superposition of the product of the singles (noncoincidence) photoelectron spectral intensity at a fixed photoelectron KE, the branching ratio of the unrenormalized decay rate of the core-hole state, and the density of final states. The coincidence Auger peak width is as narrow as the final-state width. The core-hole lifetime broadening does not contribute to the coincidence AES peak width [13,14]. When the RAES spectral intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS spectrum and the branching ratio of the renormalized (spectator or participant) decay rate of a selected decay channel, i.e., a
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partial XAS spectrum: dσ Γi = Io (ω) s dω Γs
(5)
Here Γsi is the renormalized decay rate for decay channel i, i.e., the product of the unrenormalized decay rate and the density of final states, integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel. Γsi will be asssumed to be photon energy independent. This is valid in the present case. When the renormalized decay rate depends on photon energy and it does not dominate in the total decay rate, the difference in the line shape between the XAS spectrum and the partial one manifests the photon energydependence of the renormalized decay rate. 2.2. Resonant excitation to unoccupied delocalized states Even when the core-level electron is excited to unoccupied delocalized states, if there is a mechanism by which the excited electron can transit to an unoccupied localized state lying below the unoccupied delocalized states before the core-hole decays, the resonant Auger decay will be retrived above the absorption edge. Jackson et al. [12] argued that there is little mixing between the localized state in the Si delta layer and the GaAs conduction band so that the transition between them is weak. However, the resonant core-level electron excitation probes the partial density of unoccupied states at the core-hole site. Thus, we can expect that the transition to the localized state in the Si delta layer from the conduction band at the Si site is not weak. The electron excited to the conduction band then either delocalizes from the excited core-hole site leaving the ionized Si to decay by a normal Auger decay (Fig. 2(a)), or drops into a state localized at the Si delta layer before the core-hole decays (Fig. 2(b) and (c)). The RAES spectrum will then have an additional spectral feature by resonant Auger decay of a stable excited core-hole state in which the excited electron is in a state localized at the Si delta layer. 2.2.1. Normal Auger decay First we consider the core-level electron excitation to unoccupied band states and subsequent normal Auger decay. By extending an unified theory (one-step model) of photoionization and core-hole decay, which the author developed in Ref. [14] using an unitary expansion method, we obtain the RAES spectrum (Fig. 2(a)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)Vaf (εA )|2 = dω dεA f
× Af (ε + εA − ω) dε
(6)
P˜ εc is the dipole transition matrix element for the core-level electron excitation to the unoccupied delocalized states; ε the excited electron energy; ρ the partial density of unoccupied band states; f G the core-hole propagator; Va the Auger-transition matrix element and Af is the density of final states broadened by their hole lifetime.
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
Fig. 2. (a) The normal Auger decay: the electron excited to the conduction band delocalizes rapidly, leaving the ionized atom to decay by a normal Auger decay. The core-hole propagator is dressed by the core-hole self-energy by the radiationless and radiation core-hole decays, and the relaxation rate from the metastable excited core-hole state to the stable one. (b) The resonant Auger decay: the electron excited to the conduction band drops to a state localized in the delta layer before the core-hole decays. The stable excited core-hole state decays by the radiationless participant decay. Before the metastable excited core-hole state relaxes to the stable one, the core-hole propagator is dressed in the same manner as that in (a), however, after the metastable excited core-hole state relaxes to the stable one it is dressed by the resonant radiationless and radiation decay as in Fig. 1. (c) The same process as the one in (b) except for that the stable excited core-hole state decays by radiationless spectator decay.
The core-hole propagator G is G(ε − ω) = (ε − ω − εc − i(Γ + ∆))−1
(7)
Here Γ is the core-hole lifetime width by radiationless and radiation decays. ∆ is the renormalized hopping rate of the excited electron in the delocalized states to an unoccupied localized state, i.e., the relaxation rate of a metastable excited corehole state to a stable one. The renormalized relaxation (hopping) rate is (ω) = Im |VR ( R )|2 Gca (ω − R ) d R (8) Im is the imaginary part; VR the hopping matrix element of the excited electron in the delocalized states to an unoccupied localized state and R is the relaxation energy. Eq. (6) becomes d2 σ = dω dεA
f
|P˜ εc |2 ρ(ε)Ac (ε − ω)
f
× Af (ε + εA − ω) dε
π|Va (εA )|2 (Γ + ∆) (9)
Here Ac is the core-hole spectral function, i.e., the imaginary part of Eq. (7) divided by π. The RAES spectrum in the continuous like radiationless resonant Raman scattering regime is a superposition of a convolution
of the partial density of unoccupied states, the core-hole spectral function, and the density of final states, weighted by the branching ratio of the partial core-hole decay rate. The normal Auger decay spectrum is modulated by the partial density of unoccupied states. When the band width of the unoccupied states is much smaller than the core-hole lifetime width, i.e., the time scale of delocalization of the excited electron is much longer than that of core-hole decay, the partial density of unoccupied states in Eq. (9) can be approximated by a delta function. Eq. (3) will then be retrieved so that above the absorption edge the excess energy of the photon will be transferred completely to the Auger electron and the resonant Auger decay peak appears at the constant binding energy. When the band width of the unoccupied states is much larger than the core-hole lifetime width, i.e., the time scale of delocalization of the excited electron is much shorter than that of core-hole decay, above the absorption edge the excess energy of the photon will be absorbed by the system probed so that the normal Auger decay peak appears at constant KE, however, the line shape is still modulated by the partial density of unoccupied states [1,2]. When the normal Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
spectrum and the branching ratio of the renormalized Augertransition rate of a selected decay channel. The spectrum is a partial XAS spectrum: dσ Γi = I(ω) dω (Γ + ∆)
(10)
Here Γ i is the renormalized Auger-transition rate for decay channel i, i.e., the product of the unrenormalized Augertransition rate and the density of final states, integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel and Γ i is assumed to be energy independent. The assumption is valid in the present case. I(ω) is the XAS spectrum for the resonant core-level electron excitation to the unoccupied delocalized states: I(ω) = |P˜ εc |2 ρ(ε)Ac (ε − ω) dε (11) In general when the branching ratio of the renormalized Auger-transition rate depends on the hole energy = photoelectron energy − photon energy, the difference in the line shape between the XAS spectrum and the partial one manifests the energy-dependence of the branching ratio. For an example when the core-hole lifetime is dominantly governed by a rapid core-hole decay channel such as super Coster–Kroning transition, the normal Auger decay intensity measured as a function of photon energy for a decay channel the decay rate of which is much smaller than the sCK-transition rate, the XAS spectrum is a partial one modulated by the branching ratio of the selected decay channel. The difference in the line shape between the XAS spectrum and the partial one then manifests the energydependence of the sCK-transition rate. When the final-state width is very small (i.e., the density of final states can be approximated by a delta function), and the photon width and the width of the electron energy analyzer are smaller than the hole lifetime width, the normal Auger decay intensity at a selected Auger-electron KE measured as a function of photon energy is the product of the partial density of unoccupied states at the photoelectron energy (which is fixed by the energy conservation law) and the branching ratio of the Auger-transition rate at a selected Auger-electron KE. The spectrum is a partial XAS spectrum without the core-hole lifetime broadening [6,15]. When the branching ratio is energy independent, the partial XAS spectral line shape is the partial density of unoccupied states probed by the resonant core-level electron excitation. When the branching ratio is energy-dependent as in Eq. (9), the partial XAS spectral line shape is the partial density of unoccupied states modulated by the energy-dependent branching ratio. The renormalized relaxation rate depends on photon energy so that the partial XAS spectral line shape does not reflect the partial density of unoccupied states. 2.2.2. Resonant Auger decay When the electron excited to the unoccupied delocalized states transits to an unoccupied state localized at the delta layer before the core-hole decays, the resonant AES spectrum by the resonant Auger decay of the stable excited core-hole state is
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(Fig. 2(b) and (c)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)VR ( R )Gca (ω − R ) = dω dεA f
× Vsf (εA )|2 Af (εA − ω + R )d R dε
(12)
Eq. (12) becomes d2 σ π|VR ( R )|2 = I(ω) Aca (ω − R ) dω dεA (Γ + ∆) f
f
π|Vs (εA )|2 × Af (εA − ω + R ) d R Γs
(13)
The RAES (resonant Auger decay) spectrum is a superposition of the product of the XAS spectral intensity at a selected photon energy and a convolution of the branching ratio of the unrenormalized relaxation rate, the spectral function of the stable excited core-hole state, and the density of final states, weighted by the branching ratio of the unrenormalized (participant or spectator) decay rate. The resonant Auger peak appears at a constant KE as in the case of normal Auger decay. When the effect of the energy-dependent unrenormalized relaxation rate on the spectrum is small, the resonant Auger peak width is the sum of the width of the stable excited core-hole state and the final-state width. When the resonant Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS spectrum of the metastable excited core-hole state, the branching ratio of the renormalized relaxation rate and the branching ratio of the renormalized participant or spectator decay rate of a selected decay channel. The spectrum is a partial XAS spectrum: i dσ ∆ Γs = I(ω) dω ∆ + Γ Γs
(14)
Here we assumed that the renormalized participant or spectator decay rate is energy independent. The assumption is valid in the present case. The difference in the line shape between the XAS spectrum and the partial one is due to the photon energydependent renormalized relaxation rate. When the final-state width is very small, i.e., the density of final states can be approximated by a delta function, and the photon width and the width of the electron energy analyzer are smaller than the hole lifetime width, the resonant Auger decay intensity at a selected Auger-electron KE measured as a function of photon energy is the XAS spectrum of the metastable excited core-hole state weighted by the branching ratio of the unrenormalized relaxation rate, the maximum intensity of the spectral function of the stable excited core-hole state, and the branching ratio of the renormalized decay rate. The branching ratios are evaluated for a selected Auger-electron KE.
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
3. Si K-LL RAES spectra 3.1. Resonant and normal Auger peak intensity 3.1.1. Resonant Auger peak intensity Jackson et al. [12] fitted the Si K-LL RAES spectra to two Lorentzian line shapes of variable widths and intensity convoluted with the Gaussian contribution by the analyzer. The variation with photon energy of the peak intensity is shown in Fig. 4(a) in Ref. [12]. Eq. (5) shows that when the core-level electron is excited to an empty localized state in the delta layer, the resonant Auger peak intensity measured as a function of photon energy is a partial first XAS spectrum, i.e., the product of the XAS spectrum of the stable excited core-hole state and the branching ratio of the renormalized spectator decay rate of the stable one. Eq. (14) shows that when the core-level electron is excited to empty delocalized states and the electron transits to an empty localized state before the core-hole decays, the resonant Auger peak intensity measured as a function of photon energy is a partial second XAS spectrum, i.e., the product of the XAS spectrum of the metastable excited core-hole state, the branching ratio of the renormalized relaxation rate of the metastable one and the branching ratio of the renormalized spectator decay rate of the stable one. The variation with photon energy of the resonant Auger peak intensity is then a superposition of the first XAS spectrum and the second XAS one, weighted by the respective branching ratio. Fig. 4(a) in Ref. [12] shows that in the first XAS peak region the variation with photon energy of the resonant Auger peak intensity looks like that of the first XAS spectrum, however, in the second XAS peak region the variation indicates the presence of the second XAS spectrum modulated by the photon energy-dependent branching ratio of the renormalized relaxation rate of the metastable excited core-hole state. 3.1.2. Normal Auger peak intensity Eq. (10) shows that the normal Auger peak intensity measured as a function of photon energy in the second XAS peak region is a partial second XAS spectrum, i.e., the product of the XAS spectum of the metastable excited core-hole state and the branching ratio of the renormalized normal Auger-decay rate of the metastable one. The branching ratio depends on photon energy because the renormalized relaxation rate depends on photon energy. Thus, the variation with photon energy of the normal Auger peak intensity differs from that of the second XAS spectrum. Below the threshold due to a large photon width, the electron will be partly excited to the bottom of the conduction band so that it delocalizes from the excitation site leaving the ionized Si to decay by a normal Auger decay [12]. This results in an increase in the normal Auger peak intensity with photon energy. In the threshold region a high resolution spectroscopy may show the radiationless resonant Raman scattering effect on the normal AES spectrum. 3.1.3. Relaxation rate We determine the variation with photon energy of the renormalized relaxation rate from that of the ratio (R) of the resonant Auger peak intensity to the normal one in the second XAS peak
Fig. 3. The black circle is the variation with photon energy of the relaxation rate of the metastable excited core-hole state to the stable one determined from that of the ratio of the resonant Auger peak intensity to the normal one in the photon energy of 1839.5 eV (the bottom of the conduction band) to 1841.0 eV (the maximum of the conduction band intensity). The black triangle is the variation with photon energy of the difference between the normal Auger peak width and the one at far above the ionization threshold. The difference is the relaxation width (rate). The relaxation rate determined from the ratio of the resonant Auger peak intensity to the normal one is less accurate, compared to that from the width difference. As discussed in the text the resonant or normal Auger peak intensity is less accurate because of a large photon width. When the accuracy is taken into account, the agreement between the two set of the relaxation rate is reasonably good. Note that the relaxation width and the peak width difference are defined as a half width at the half maximum (HWHM).
region. We assume that the branching ratio of the K-LL spectator decay rate is the same as that of the normal K-LL Auger one. As the Auger-electron KE is as high as 1600 eV, the presence of a spectator electron in the localized state will not affect the transition rate. The participant decay rate is also assumed to be small. The ratio (R) is then R=
∆ Γ
(15)
The lifetime width of 1s hole in Si is 0.38 eV [12,16], while the theoretical one is 0.425 eV [17]. In Fig. 3 we show the renormalized relaxation rate determined by Eq. (15). The relaxation rate decreases with an increase in photon energy. The relaxation rate maximizes at the bottom of the conduction band. If we can describe the relaxation rate as the product of the coupling constant and the partial density of unoccupied states, the photon energy-dependence of the relaxation rate reflects the partial density of unoccupied states. 4. RAES peak width 4.1. Resonant Auger peak width The variation with photon energy of the resonant or normal Auger peak width is shown in Fig. 5 in Ref. [12]. The resonant Auger peak width approaches and merges the normal one with increasing photon energy. Eq. (3) shows that the resonant Auger peak width in the first XAS peak region is the
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
final-state width (0.5–0.6 eV), whereas if the effect of the energydependent unrenormalized relaxation rate is negligible, Eq. (13) shows that the resonant Auger peak width in the second XAS peak region is the sum of the lifetime width of the stable resonantly excited core-hole state and the final-state width, i.e., the sum of the spectator decay width, the participant decay width, and the final-state width. The core-hole lifetime of the stable resonantly excited core-hole state is approximately the same as that of the ionized core-hole state. Note that under this assumption the relaxation rate is determined. Fig. 5 in Ref. [12] shows that the resonant Auger peak width in the second XAS peak region is approximately equal to the normal one measured at far above the threshold. The latter width is the sum of the core-hole lifetime width and the final-state width. The resonant Auger peak width increases linearly from the final-state width to the sum of the core-hole lifetime width and the final-state width. This is because in the transition region from the first XAS peak to the second XAS peak the resonant Auger decay spectrum is a superposition of that of the stable excited core-hole state and that of the metastable one which relaxes to the stable one before the core-hole decays. The latter spectrum arises because of a large photon width. The presence of the latter spectrum affects the variation with photon energy of the resonant Auger peak KE in the same region (see later). 4.2. Normal Auger peak width The normal Auger peak width (0.9 eV [12]) at 33 eV above the ionization threshold is the sum of the core-hole lifetime width (0.38 eV [12,16]) and the final-state width. The final-state width is then 0.52 eV. It agrees well with the resonant Auger peak width (0.5 eV) in the first XAS peak region, which is the finalstate width. Eq. (9) shows that the normal Auger peak width in the second XAS peak region is the sum of the core-hole lifetime width (0.38 eV), the final-state width (0.5 eV), and the relaxation rate. In Fig. 3 we plot the variation with photon energy of the difference between the normal Auger peak width at 1873 eV and the one in the second XAS peak region. The variation with photon energy of the width difference agrees qualitatively with that of the relaxation rate. The deviations are within the experimental accuracy of the intensity ratio of the resonant Auger peak to the normal one limited by a large photon width (Fig. 4(a) in Ref. [12]). As the photon width is not small, the normal Auger decay in the first XAS peak region is mostly that of the core-hole state in which the electron excited to the bottom of the conduction band is delocalized. The normal Auger peak width then becomes the same as that at 1839.5 eV. The present scenario can describe the variation with photon energy of the normal Auger peak width. 5. RAES peak KE 5.1. Resonant Auger peak KE The variation with photon energy of the resonant or normal Auger peak KE is shown in Fig. 6 in [12]. The resonant Auger peak KE shows no dispersion with photon energy except when
19
excited by the highest energy photons. The resonant Auger peak KE shows a constant KE in contrast to a positive linear dispersion with photon energy shown by most other resonant Auger decays. Eq. (13) shows that the resonant Auger peak KE in the second XAS peak region is independent of photon energy and is the same as that at the first XAS peak (1839 eV). In the first XAS peak region the resonant Auger peak KE should show a positive linear dispersion with photon energy. However, it shows almost no dispersion. This is because of the resonant Auger decay of the metastable excited core-hole state which relaxes to the stable one before the core-hole decays. Because of a large photon width the presence of the resonant Auger decay of the metastable core-hole state increases with photon energy. Thus, with an increase in photon energy a resonant Auger peak of constant KE becomes more dominant than that of a positive linear dispersion. This results also in an increase in the resonant Auger peak width and intensity in the same photon energy region. The present scenario explains why the resonant Auger peak KE does not show any dispersion with photon energy. The variation with photon energy of the resonant Auger peak KE shows that at 1843 eV, i.e., 2 eV above the second XAS peak, the KE increases by about 2 eV compared to that at the second XAS peak (1841.1 eV) [12]. According to the scenario by Jackson et al. [12], the resonant Auger peak KE at 1843 eV should increase by 4.2 eV (1843–1838.8 eV) compared to that at 1838.8 eV because the resonant Auger peak KE is expected to follow a positive linear dispersion. The relative resonant Auger peak intensity in the RAES spectrum at 1843 eV is very small and the normal Auger peak intensity dominates (Fig. 4(a) in Ref. [12]). Above the second XAS peak energy (1841.1 eV) where the relaxation rate diminishes, there should be another mechanism by which the resonant Auger peak appears. The KE difference between the resonant Auger peak and the normal one is 2Uc a − Uca. Here Uca is the screened Coulomb hole–particle interaction energy in the stable excited core-hole state, i.e., the one between the 1s hole and the excited electron in the localized state, while Uc a is the one in the final state of the spectator decay of the stable excited core-hole state, i.e., the one between the single 2p hole and the excited electron in the localized state. The normal Auger peak KE is 1607 eV [12], whereas the resonant one is 1611 eV [12] (2Uc a − Uca = 4 eV). Thus, the potential in the two 2p hole final state is much more attractive than that in the single 1s hole state. At 1843 eV the normal Auger decay spectrum dominates in the RAES spectrum. Thus, the excited electron in the conduction band is most likely to delocalize before the core-hole decays. As the relaxation rate diminishes above the second XAS peak (Fig. 3), there is still a possibility that the excited electron remains in the conduction band during the core-hole decay. The excited electron will then be exposed to two holes created by the core-hole decay. As the two-hole potential is much more attractive than the single-hole one, the excited electron is pulled down to the maximum of the conduction band. The excited electron is then further pulled down to the localized state in the delta layer (Fig. 4(a)). Above the second XAS peak the resonant Auger
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
Fig. 4. (a) The electron excited at 1843 eV drops to the maximum of the conduction band by a change from a single core-hole potential to a more attractive twohole potential created by the Auger transition of the ionized state. The excited electron then drops to a state localized in the delta layer. (b) The PCI between the photoelectron and the Auger electron emitted. The photoelectron will lose its KE by a change from a single core-hole potential to a more attractive two-hole potential created by the normal Auger decay of the core-hole state. At the same time the Auger electron emitted gains the KE by the screening of one of the two holes by the photoelectron.
peak KE then becomes εA = ω + ε f + ω a − ω b
(16)
Here εf is the 2h1p final-state energy and ωb is the second XAS peak energy (1841.1 eV), while ωa is the first one (1838.8 eV). Eq. (16) shows that above the second XAS line peak the resonant Auger KE shows a positive linear dispersion with photon energy (Fig. 6 in Ref. [12]). The present scenario can explain well the variation with photon energy of the resonant Auger peak KE. 5.2. Normal Auger peak KE The normal Auger peak KE decreases from the first XAS peak region to the second one. Because of a large photon width the excitation to the bottom of the conduction band becomes possible. If the radiationless Raman scattering plays an important role, the normal Auger peak KE in the threshold region should increase and approach the one at far above the threshold with photon energy. However, the variation with photon energy of the normal Auger peak KE is opposite. This indicates the presence of the PCI effect [12]. The excited electron acts as a slow photoelectron. The slow photoelectron recedes from the ionic atomic site experiencing an attractive Coulomb core-hole potential. A fast Auger electron is then emitted as the core-hole decays. If the lifetime of the core-hole is sufficiently short, the fast Auger electron can overtake the photoelectron, which is then exposed to a doubly charged ion core. As the doubly charged ion core potential becomes more attractive than the core-hole potential, the photoelectron will be retarded losing energy, whereas the Auger electron now exposed to a single charged ion core screened by the slow photoelectron, gains energy. The PCI is caused by a sudden change of the Coulomb field that a fast Auger electron experiences on overtaking an initially ejected slow photoelectron [18,19]. When the PCI effect is considered, the normal Auger decay spectrum becomes (Fig. 4(b)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)Vaf (εA )ε|˜ε|2 = dω dεA f
× δ(˜ε − εf − ω + εA ) dε d˜ε
(17)
Eq. (17) becomes d2 σ |P˜ εc |2 ρ(ε)Ac (ε − ω) = dω dεA f
f
×
π|Va (εA )|2 |ε|ω + εf − εA |2 dε Γ +∆
(18)
The normal Auger decay spectrum is modulated by |ε|˜ε|2 where | is the photoelectron wavefunction in the singly charged ion core potential, while |˜ε is the one in the doubly charged ion core potential. When the normal Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is a partial XAS spectrum modulated by the energy-dependent branching ratio of the Auger-transition rate of a selected decay channel. 6. Conclusion We discussed the RAES spectra of Si delta dopped layers in GaAs with very thin capping layers using a many-body theory. The resonant Auger peak KE shows no dispersion with photon energy, except when excited by the highest energy photons, in contrast to a positive dispersion with photon energy shown by most other resonant Auger decays. This is explained in terms of the relaxation of the metastable excited core-hole state to the stable one before the core-hole decay. The electron excited to the conduction band either delocalizes leaving the ionized Si to decay by a normal Auger decay or transits to a state localized in the delta layer before the core-hole decays. This results in the presence of both normal Auger decay and resonant one, when the photon energy exceeds the one for the excitation to the bottom of the conduction band. The resonant Auger peak KE is the same as that at the resonant excitation to the empty localized state in the delta layer. The resonant Auger peak width is the sum of the corehole lifetime of the stable excitated core-hole state (the excited electron is in the localized state in the delta layer) and the finalstate width. Thus, the resonant Auger peak width becomes larger
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
than the one at the resonant excitation to the stable excited corehole state. The former width is approximately the same as the normal Auger peak width measured at far above the threshold. The photon energy-dependent relaxation rate is determined from the variation with photon energy of the intensity ratio of the resonant Auger peak to the normal one in the photon energy region in which the electron is excited to the conduction band. The relaxation rate which maximizes at the bottom of the conduction band decreases with photon energy. The variation with photon energy of normal Auger peak width in the photon energy region of the resonant excitation to the conduction band can be explained in terms of that of the relaxation rate. Above the metastable (second) XAS peak the resonant Auger KE shows a positive linear dispersion with photon energy. This is explained in terms of the relaxation of the excited electron in the conduction band to a localized state in the delta layer by an attractive two-hole potential in the final state of core-hole decay. A low resolution of the measurement (a large photon width) explains an increase of the resonant Auger peak width with photon energy and a constant resonant Auger peak KE below the threshold. The present theory is applicable to the case in which a similar relaxation from a metastable excited core-hole state to a stable one on time scale of core-hole decay occurs. A high resolution measurement is desirable. References ˚ [1] T. Aberg, Phys. Scr. T41 (1992) 71. ˚ [2] T. Aberg, B. Crasemann, in: G. Materlik, C.J. Sparks, K. Fischer (Eds.), Resonant Anomalous X-ray Scattering, Elsevier Science, NY, 1994, p. 431.
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[3] G.S. Brown, M.H. Chen, B. Crasemann, G.E. Ice, Phys. Rev. Lett. 45 (1980) 1937. [4] W. Drube, R. Treusch, R. Dahn, M. Greibenow, M. Grehk, G. Materlik, J. Electron Spectrosc. Relat. Phenom. 79 (1996) 223. [5] M. Elango, A. Ausmees, A. Kikas, A. Maiste, E. Nommiset, A. Saar, J. Electron Spectrosc. Relat. Phenom. 72 (1995) 127. [6] W. Drube, R. Treusch, G. Materlik, Phys. Rev. Lett. 74 (1995) 42. [7] O. Bj¨orneholm, A. Nilsson, A. Sandell, B. Hernn¨as, N. M˚artensson, Phys. Rev. Lett. 68 (1992) 1892. [8] W. Wurth, P. Feulner, D. Menzel, Phys. Scr. T 41 (1992) 213. [9] M. Ohno, Phys. Rev. B 50 (1994) 2566. [10] O. Karis, A. Nilsson, M. Weinelt, T. Wiell, C. Puglia, N. Wassdahl, N. M˚artensson, M. Samant, J. St¨ohr, Phys. Rev. Lett. 76 (1996) 1380. [11] C. Keller, M. Stichler, G. Comelli, F. Esch, S. Lizzit, D. Menzel, W. Wurth, Phys. Rev. B 57 (1998) 11951. [12] M.D. Jackson, J.M.C. Thornton, D. Lewis, A. Robinson, M. Fahy, A. Aviary, P. Weightman, Phys. Rev. B 71 (2005) 075313, and references therein. [13] H.W. Haak, G.A. Sawatzky, T.D. Thomas, Phys. Rev. Lett. 41 (1978) 1825. [14] M. Ohno, G. Wendin, J. Phys. B 12 (1979) 1305. [15] W. Drube, A. Lessmann, G. Materlik, Jpn. J. Appl. Phys. 32 (Suppl. 32-2) (1993) 173. [16] P.S. Fowles, J.A. Evans, P.M. Lee, A.D. Laine, P. Weightman, Phys. Rev. B 48 (1993) 14142. [17] S.T. Perkins, D.E. Cullen, M.H. Chen, J.H. Hubbell, L. Rathkopf, J. Scofield, Lawrence Livemore National Laboratory, Evaluated Atomic Data Library, UCRL-50400, vol. 30, 1991. [18] A. Niehaus, J. Phys. B 10 (1977) 1845. [19] M.Yu. Kuchiev, S.A. Sheinerman, Sov. Phys. JETP 63 (1986) 986.
Many-electron effect in the Si K-LL resonant Auger-electron spectroscopy spectra of the Si delta layer in GaAs Masahide Ohno Quantum Science Research, 2-8-5 Tokiwadai, Itabashi-ku, Tokyo 174-0071, Japan Received 6 December 2005; received in revised form 6 April 2006; accepted 10 April 2006 Available online 5 July 2006
Abstract The Si K-LL resonant Auger-electron spectroscopy (RAES) spectra of silicon delta dopped layers in GaAs with very thin capping layers show both normal Auger decay and resonant Auger decay, when the core-level electron is excited to the conduction band. The resonant Auger peak kinetic energy (KE) shows no dispersion with photon energy, except when excited by the highest energy photons [M.D. Jackson, J.M.C. Thornton, D. Lewis, A. Robinson, M. Fahy, A. Aviary, P. Weightman, Phys. Rev. B71 (2005) 075313]. The RAES spectra are analyzed using a many-body theory. The presence of resonant Auger decay and no dispersion of resonant Auger peak KE with photon energy is explained in terms of the relaxation of a metastable excited core-hole state to a stable one on the time scale of core-hole decay. The excited electron in the conduction band either delocalizes rapidly leaving the ionized Si to decay by a normal Auger decay or drops to a state localized in the Si delta layer before the core-hole decays so that the RAES spectrum has both normal Auger decay and resonant Auger decay. As a result of the relaxation, the resonant Auger peak KE does not show any dispersion with photon energy. The variations with photon energy of the normal or resonant Auger peak intensity, KE, and width are explained in a consistent manner by a many-body theory. © 2006 Elsevier B.V. All rights reserved. Keywords: Si delta dopped layers in GaAs; Resonant Auger-electron spectroscopy (RAES); Normal Auger decay; Resonant Auger decay; Resonant radiationless Raman scattering; Core-hole lifetime; Core-hole relaxation
1. Introduction As a result of energy conservation of a coherent one-step process of resonant excitation, autoionization decay channels show two distinct characteristic resonant Auger–Raman effects, i.e., linear dispersion of the kinetic energy (KE) of the decay channels with excitation energy and line narrowing [1–6]. When the excited electron is present during the core-hole decay, the decay spectra show the resonant Auger–Raman effects, while those where the excited electron delocalizes before the decay will show normal Auger decay [7–11]. When the core-level electron is excited to unoccupied states the bandwidth of which is much smaller than the core-hole lifetime width, above the absorption edge the excess energy of the photon will be transferred completely to the Auger electron so that the resonant Auger-electron spectroscopy (RAES) line appears at the constant binding energy. However, when the core-level electron is excited to unoccupied states the bandwidth of which is much larger than the core-hole lifetime width, above the absorption edge the excess energy of the photon will be absorbed by the 0368-2048/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2006.04.001
system probed so that the RAES line appears at constant KE, however, the line shape is still modulated by the partial density of unoccupied states. Even when the core-level electron is excited into unoccupied delocalized states, if there is a mechanism by which the excited electron can transit to an unoccupied localized state lying below the unoccupied band states on the time scale of core-hole decay, the resonant Auger decay will be retrived even above the absorption edge, in addition to the normal Auger decay. Recently Jackson et al. [12] measured the K X-ray absorption spectroscopy (XAS) spectra and the RAES spectra of silicon delta dopped layers in GaAs with very thin capping layers. In the present paper we focus on the RAES spectra of the 0.01 ML sample. The Si K-edge XAS spectrum shows two distinct features: a pre-edge to the absorption which occurs at an excitation energy of 1838.8 eV, and a second peak at about 1841 eV (Fig. 2(a) in Ref. [12]). The former XAS peak arises from a localized Si state at the Si layer and does not involve the electronic structure of the GaAs. The second peak arises from the maximum intensity of the conduction band of GaAs which occurs
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at 1.4 eV above the bottom of the conduction band (=1839.7 eV [12]). The energy of the second peak corresponds to the Si 1s ionization energy. When excited far above threshold the Si KL23 L23 (=KLL) AES spectra show the characteristic multiplet structure which consists of a strong component arising from the 1 D term of the two 2p hole final state which was observed at 2 a KE of 1606 eV and a weak component which arises from the 1 S term and occurs 6 eV lower in KE. When excited by photons close to threshold the Si KLL RAES spectra do not show this characteristic multiplet structure and vary markedly with photon energy (Fig. 3(a) in Ref. [12]). The spectrum consists of two main peaks and of these the one to lower KE evolves into the D term observed at a constant KE of 1606 eV when excited by photons far above threshold. They associate the peak to higher KE with a resonant Auger decay as it is observed when excited by radiation close to the K threshold and it exhibits narrower width than the D term. The maximum intensity of this feature is observed when the spectrum is excited by photons corresponding to the maximum of the first peak in the XAS spectrum, i.e., the Si 1s electron is excited into a localized state associated with the delta layer (Fig. 4(a) in Ref. [12]). The resonant Auger decay requires a long-lived localized excited state, i.e., the characteristic of a state localized at a delta layer in the band gap of a semiconductor [12]. The resonant Auger peak remains at a constant KE (1611 eV, the same as the one measured at the first XAS line peak (1838.8 eV)) over the 2 eV photon excitation range (1839–1841 eV) for which it is intense (Figs. 3(a) and 6 in Ref. [12]). The resonant Auger peak shows no dispersion with photon energy except when excited by the highest energy photons. This is in contrast to most other resonant Auger decays, which show a positive dispersion with photon energy ([3–6] and references in [12]). Jackson et al. [12] suggested that a likely explanation for this behavior is that in the delta layer the unoccupied state is very narrow and shows no dispersion in energy. With these characteristics it can only be excited by a narrow range of photon energies, a range that may be narrower than the resolution of their measurement. They pointed out that at higher photon
energies it is possible that the localized state can be occupied by electrons, which are first excited into the GaAs conduction band and which then transfer their energy to the resonant Auger electron when dropping into the localized state. While this would provide a mechanism to increase the resonant Auger-electron KE, as exhibited in the spectrum excited by the highest energy photons (Fig. 6 in Ref. [12]), they argued that there will be little mixing between the localized state in the Si delta layer and the GaAs conduction band, and the transition will be weak [12]. The electron excited to the bottom of the GaAs conduction band would be expected to delocalize from the excitation site leaving the ionized Si to decay by a normal Auger decay [12]. The normal Auger peak KE shows a negative dispersion with photon energy (Fig. 6 in Ref. [12]), which could be due to the postcollision interaction (PCI) effect [12]. The resonant Auger peak width shows an increase with photon energy, whereas the normal Auger peak width shows an opposite trend. Both widths merge at the second XAS peak energy (Fig. 5 in Ref. [12]). The variation with photon energy of the resonant or normal Auger peak intensity is shown in Fig. 4 in Ref. [12] The variations with photon energy of the resonant or normal Auger peak KE, width and intensity require an explanation. In the present paper by a many-body theory we shall discuss the RAES spectral behavior in terms of the transition of an excited electron in delocalized states to a localized state on time scale of core-hole decay. 2. Theory 2.1. Resonant excitation to a localized state and resonant Auger decay We consider the resonant core-level electron excitation to an unoccupied localized state and subsequent resonant Auger decay. The RAES spectrum is (Fig. 1): d2 σ = |P˜ ca |2 |Gca (ω)Vsf (εA )|2 Af (εA − ω) dω dεA
(1)
f
Fig. 1. The resonant Auger decay: (a) The radiationless participant decay of the resonantly excited core-hole state in which the excited electron occupies an unoccupied localized state such as the one in the Si delta layer. The core-hole propagator of the excited core-hole state is dressed by the core-hole self-energy by both radiationless and radiation decay. (b) The radiationless spectator decay of the resonantly excited core-hole state defined in the same manner as in (a). The excited electron remains localized as a spectator during the core-hole decay. When the screening of the two-hole potential in the final state by the spectator electron is effective, and the resonant Auger-electron KE is small, the presence of the spectator electron affects the resonant Auger decay rate so that the term “spectator” is not appropriate anymore. This is not the case in the present case.
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
P˜ ca is the dipole-transition matrix element of the core-level electron excitation to an unoccupied localized state; ω the incident photon energy; Gca the core-hole excited-electron pair (or the f excited core-hole state) propagator; Vs the radiationless participant or spectator decay matrix element of the excited core-hole state; εA the KE of the electron emitted by the core-hole decay and Af is the density (or the spectral function) of the final states broadened by their hole lifetime. The final state by the radiationless participant decay is a single-hole (1h) state, while that by the radiationless spectator decay is a two-hole one-particle (2h1p) state. The photon energy distribution and the spectral function of the electron analyzer can be additionally included in Eq. (1). The core-hole excited-electron pair propagator Gca is Gca (ω) = (εa − εc − ω − iΓs )−1
(2)
Here εa is the energy of an unoccupied localized state in the presence of the core-hole; εc the core-hole energy and Γ s is the lifetime width of the excited core-hole state by the radiation and radiationless decays. Eq. (1) becomes f
d2 σ π|Vs (εA )|2 = Io (ω) Af (εA − ω) dω dεA Γs
(3)
f
Io (ω) is the XAS spectrum for the resonant core-level electron excitation to an unoccupied localized state. Io (ω) = |P˜ ca |2 Aca (ω)
(4)
Aca is the spectral function of the resonantly excited core-hole state, i.e., the imaginary part of the core-hole excited-electron pair (or the excited core-hole state) propagator (Eq. (2)) divided by π. The RAES (resonant Auger decay) spectrum is a superposition of the product of the XAS spectral intensity at a selected photon energy, the branching ratio of the unrenormalized radiationless (spectator or participant) decay rate of the excited core-hole state, and the density (the spectral function) of the 1h or 2h1p final states. The RAES peak appears at the constant binding energy. The RAES peak KE shows a positive dispersion with photon energy. The RAES peak width is as narrow as the final-state width. The peak width narrowing is analogous to that of the normal Auger peak, when the Auger electrons are collected in coincidence with photoelectrons of a fixed KE. The coincidence AES spectrum is a superposition of the product of the singles (noncoincidence) photoelectron spectral intensity at a fixed photoelectron KE, the branching ratio of the unrenormalized decay rate of the core-hole state, and the density of final states. The coincidence Auger peak width is as narrow as the final-state width. The core-hole lifetime broadening does not contribute to the coincidence AES peak width [13,14]. When the RAES spectral intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS spectrum and the branching ratio of the renormalized (spectator or participant) decay rate of a selected decay channel, i.e., a
15
partial XAS spectrum: dσ Γi = Io (ω) s dω Γs
(5)
Here Γsi is the renormalized decay rate for decay channel i, i.e., the product of the unrenormalized decay rate and the density of final states, integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel. Γsi will be asssumed to be photon energy independent. This is valid in the present case. When the renormalized decay rate depends on photon energy and it does not dominate in the total decay rate, the difference in the line shape between the XAS spectrum and the partial one manifests the photon energydependence of the renormalized decay rate. 2.2. Resonant excitation to unoccupied delocalized states Even when the core-level electron is excited to unoccupied delocalized states, if there is a mechanism by which the excited electron can transit to an unoccupied localized state lying below the unoccupied delocalized states before the core-hole decays, the resonant Auger decay will be retrived above the absorption edge. Jackson et al. [12] argued that there is little mixing between the localized state in the Si delta layer and the GaAs conduction band so that the transition between them is weak. However, the resonant core-level electron excitation probes the partial density of unoccupied states at the core-hole site. Thus, we can expect that the transition to the localized state in the Si delta layer from the conduction band at the Si site is not weak. The electron excited to the conduction band then either delocalizes from the excited core-hole site leaving the ionized Si to decay by a normal Auger decay (Fig. 2(a)), or drops into a state localized at the Si delta layer before the core-hole decays (Fig. 2(b) and (c)). The RAES spectrum will then have an additional spectral feature by resonant Auger decay of a stable excited core-hole state in which the excited electron is in a state localized at the Si delta layer. 2.2.1. Normal Auger decay First we consider the core-level electron excitation to unoccupied band states and subsequent normal Auger decay. By extending an unified theory (one-step model) of photoionization and core-hole decay, which the author developed in Ref. [14] using an unitary expansion method, we obtain the RAES spectrum (Fig. 2(a)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)Vaf (εA )|2 = dω dεA f
× Af (ε + εA − ω) dε
(6)
P˜ εc is the dipole transition matrix element for the core-level electron excitation to the unoccupied delocalized states; ε the excited electron energy; ρ the partial density of unoccupied band states; f G the core-hole propagator; Va the Auger-transition matrix element and Af is the density of final states broadened by their hole lifetime.
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
Fig. 2. (a) The normal Auger decay: the electron excited to the conduction band delocalizes rapidly, leaving the ionized atom to decay by a normal Auger decay. The core-hole propagator is dressed by the core-hole self-energy by the radiationless and radiation core-hole decays, and the relaxation rate from the metastable excited core-hole state to the stable one. (b) The resonant Auger decay: the electron excited to the conduction band drops to a state localized in the delta layer before the core-hole decays. The stable excited core-hole state decays by the radiationless participant decay. Before the metastable excited core-hole state relaxes to the stable one, the core-hole propagator is dressed in the same manner as that in (a), however, after the metastable excited core-hole state relaxes to the stable one it is dressed by the resonant radiationless and radiation decay as in Fig. 1. (c) The same process as the one in (b) except for that the stable excited core-hole state decays by radiationless spectator decay.
The core-hole propagator G is G(ε − ω) = (ε − ω − εc − i(Γ + ∆))−1
(7)
Here Γ is the core-hole lifetime width by radiationless and radiation decays. ∆ is the renormalized hopping rate of the excited electron in the delocalized states to an unoccupied localized state, i.e., the relaxation rate of a metastable excited corehole state to a stable one. The renormalized relaxation (hopping) rate is (ω) = Im |VR ( R )|2 Gca (ω − R ) d R (8) Im is the imaginary part; VR the hopping matrix element of the excited electron in the delocalized states to an unoccupied localized state and R is the relaxation energy. Eq. (6) becomes d2 σ = dω dεA
f
|P˜ εc |2 ρ(ε)Ac (ε − ω)
f
× Af (ε + εA − ω) dε
π|Va (εA )|2 (Γ + ∆) (9)
Here Ac is the core-hole spectral function, i.e., the imaginary part of Eq. (7) divided by π. The RAES spectrum in the continuous like radiationless resonant Raman scattering regime is a superposition of a convolution
of the partial density of unoccupied states, the core-hole spectral function, and the density of final states, weighted by the branching ratio of the partial core-hole decay rate. The normal Auger decay spectrum is modulated by the partial density of unoccupied states. When the band width of the unoccupied states is much smaller than the core-hole lifetime width, i.e., the time scale of delocalization of the excited electron is much longer than that of core-hole decay, the partial density of unoccupied states in Eq. (9) can be approximated by a delta function. Eq. (3) will then be retrieved so that above the absorption edge the excess energy of the photon will be transferred completely to the Auger electron and the resonant Auger decay peak appears at the constant binding energy. When the band width of the unoccupied states is much larger than the core-hole lifetime width, i.e., the time scale of delocalization of the excited electron is much shorter than that of core-hole decay, above the absorption edge the excess energy of the photon will be absorbed by the system probed so that the normal Auger decay peak appears at constant KE, however, the line shape is still modulated by the partial density of unoccupied states [1,2]. When the normal Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
spectrum and the branching ratio of the renormalized Augertransition rate of a selected decay channel. The spectrum is a partial XAS spectrum: dσ Γi = I(ω) dω (Γ + ∆)
(10)
Here Γ i is the renormalized Auger-transition rate for decay channel i, i.e., the product of the unrenormalized Augertransition rate and the density of final states, integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel and Γ i is assumed to be energy independent. The assumption is valid in the present case. I(ω) is the XAS spectrum for the resonant core-level electron excitation to the unoccupied delocalized states: I(ω) = |P˜ εc |2 ρ(ε)Ac (ε − ω) dε (11) In general when the branching ratio of the renormalized Auger-transition rate depends on the hole energy = photoelectron energy − photon energy, the difference in the line shape between the XAS spectrum and the partial one manifests the energy-dependence of the branching ratio. For an example when the core-hole lifetime is dominantly governed by a rapid core-hole decay channel such as super Coster–Kroning transition, the normal Auger decay intensity measured as a function of photon energy for a decay channel the decay rate of which is much smaller than the sCK-transition rate, the XAS spectrum is a partial one modulated by the branching ratio of the selected decay channel. The difference in the line shape between the XAS spectrum and the partial one then manifests the energydependence of the sCK-transition rate. When the final-state width is very small (i.e., the density of final states can be approximated by a delta function), and the photon width and the width of the electron energy analyzer are smaller than the hole lifetime width, the normal Auger decay intensity at a selected Auger-electron KE measured as a function of photon energy is the product of the partial density of unoccupied states at the photoelectron energy (which is fixed by the energy conservation law) and the branching ratio of the Auger-transition rate at a selected Auger-electron KE. The spectrum is a partial XAS spectrum without the core-hole lifetime broadening [6,15]. When the branching ratio is energy independent, the partial XAS spectral line shape is the partial density of unoccupied states probed by the resonant core-level electron excitation. When the branching ratio is energy-dependent as in Eq. (9), the partial XAS spectral line shape is the partial density of unoccupied states modulated by the energy-dependent branching ratio. The renormalized relaxation rate depends on photon energy so that the partial XAS spectral line shape does not reflect the partial density of unoccupied states. 2.2.2. Resonant Auger decay When the electron excited to the unoccupied delocalized states transits to an unoccupied state localized at the delta layer before the core-hole decays, the resonant AES spectrum by the resonant Auger decay of the stable excited core-hole state is
17
(Fig. 2(b) and (c)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)VR ( R )Gca (ω − R ) = dω dεA f
× Vsf (εA )|2 Af (εA − ω + R )d R dε
(12)
Eq. (12) becomes d2 σ π|VR ( R )|2 = I(ω) Aca (ω − R ) dω dεA (Γ + ∆) f
f
π|Vs (εA )|2 × Af (εA − ω + R ) d R Γs
(13)
The RAES (resonant Auger decay) spectrum is a superposition of the product of the XAS spectral intensity at a selected photon energy and a convolution of the branching ratio of the unrenormalized relaxation rate, the spectral function of the stable excited core-hole state, and the density of final states, weighted by the branching ratio of the unrenormalized (participant or spectator) decay rate. The resonant Auger peak appears at a constant KE as in the case of normal Auger decay. When the effect of the energy-dependent unrenormalized relaxation rate on the spectrum is small, the resonant Auger peak width is the sum of the width of the stable excited core-hole state and the final-state width. When the resonant Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is the product of the XAS spectrum of the metastable excited core-hole state, the branching ratio of the renormalized relaxation rate and the branching ratio of the renormalized participant or spectator decay rate of a selected decay channel. The spectrum is a partial XAS spectrum: i dσ ∆ Γs = I(ω) dω ∆ + Γ Γs
(14)
Here we assumed that the renormalized participant or spectator decay rate is energy independent. The assumption is valid in the present case. The difference in the line shape between the XAS spectrum and the partial one is due to the photon energydependent renormalized relaxation rate. When the final-state width is very small, i.e., the density of final states can be approximated by a delta function, and the photon width and the width of the electron energy analyzer are smaller than the hole lifetime width, the resonant Auger decay intensity at a selected Auger-electron KE measured as a function of photon energy is the XAS spectrum of the metastable excited core-hole state weighted by the branching ratio of the unrenormalized relaxation rate, the maximum intensity of the spectral function of the stable excited core-hole state, and the branching ratio of the renormalized decay rate. The branching ratios are evaluated for a selected Auger-electron KE.
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
3. Si K-LL RAES spectra 3.1. Resonant and normal Auger peak intensity 3.1.1. Resonant Auger peak intensity Jackson et al. [12] fitted the Si K-LL RAES spectra to two Lorentzian line shapes of variable widths and intensity convoluted with the Gaussian contribution by the analyzer. The variation with photon energy of the peak intensity is shown in Fig. 4(a) in Ref. [12]. Eq. (5) shows that when the core-level electron is excited to an empty localized state in the delta layer, the resonant Auger peak intensity measured as a function of photon energy is a partial first XAS spectrum, i.e., the product of the XAS spectrum of the stable excited core-hole state and the branching ratio of the renormalized spectator decay rate of the stable one. Eq. (14) shows that when the core-level electron is excited to empty delocalized states and the electron transits to an empty localized state before the core-hole decays, the resonant Auger peak intensity measured as a function of photon energy is a partial second XAS spectrum, i.e., the product of the XAS spectrum of the metastable excited core-hole state, the branching ratio of the renormalized relaxation rate of the metastable one and the branching ratio of the renormalized spectator decay rate of the stable one. The variation with photon energy of the resonant Auger peak intensity is then a superposition of the first XAS spectrum and the second XAS one, weighted by the respective branching ratio. Fig. 4(a) in Ref. [12] shows that in the first XAS peak region the variation with photon energy of the resonant Auger peak intensity looks like that of the first XAS spectrum, however, in the second XAS peak region the variation indicates the presence of the second XAS spectrum modulated by the photon energy-dependent branching ratio of the renormalized relaxation rate of the metastable excited core-hole state. 3.1.2. Normal Auger peak intensity Eq. (10) shows that the normal Auger peak intensity measured as a function of photon energy in the second XAS peak region is a partial second XAS spectrum, i.e., the product of the XAS spectum of the metastable excited core-hole state and the branching ratio of the renormalized normal Auger-decay rate of the metastable one. The branching ratio depends on photon energy because the renormalized relaxation rate depends on photon energy. Thus, the variation with photon energy of the normal Auger peak intensity differs from that of the second XAS spectrum. Below the threshold due to a large photon width, the electron will be partly excited to the bottom of the conduction band so that it delocalizes from the excitation site leaving the ionized Si to decay by a normal Auger decay [12]. This results in an increase in the normal Auger peak intensity with photon energy. In the threshold region a high resolution spectroscopy may show the radiationless resonant Raman scattering effect on the normal AES spectrum. 3.1.3. Relaxation rate We determine the variation with photon energy of the renormalized relaxation rate from that of the ratio (R) of the resonant Auger peak intensity to the normal one in the second XAS peak
Fig. 3. The black circle is the variation with photon energy of the relaxation rate of the metastable excited core-hole state to the stable one determined from that of the ratio of the resonant Auger peak intensity to the normal one in the photon energy of 1839.5 eV (the bottom of the conduction band) to 1841.0 eV (the maximum of the conduction band intensity). The black triangle is the variation with photon energy of the difference between the normal Auger peak width and the one at far above the ionization threshold. The difference is the relaxation width (rate). The relaxation rate determined from the ratio of the resonant Auger peak intensity to the normal one is less accurate, compared to that from the width difference. As discussed in the text the resonant or normal Auger peak intensity is less accurate because of a large photon width. When the accuracy is taken into account, the agreement between the two set of the relaxation rate is reasonably good. Note that the relaxation width and the peak width difference are defined as a half width at the half maximum (HWHM).
region. We assume that the branching ratio of the K-LL spectator decay rate is the same as that of the normal K-LL Auger one. As the Auger-electron KE is as high as 1600 eV, the presence of a spectator electron in the localized state will not affect the transition rate. The participant decay rate is also assumed to be small. The ratio (R) is then R=
∆ Γ
(15)
The lifetime width of 1s hole in Si is 0.38 eV [12,16], while the theoretical one is 0.425 eV [17]. In Fig. 3 we show the renormalized relaxation rate determined by Eq. (15). The relaxation rate decreases with an increase in photon energy. The relaxation rate maximizes at the bottom of the conduction band. If we can describe the relaxation rate as the product of the coupling constant and the partial density of unoccupied states, the photon energy-dependence of the relaxation rate reflects the partial density of unoccupied states. 4. RAES peak width 4.1. Resonant Auger peak width The variation with photon energy of the resonant or normal Auger peak width is shown in Fig. 5 in Ref. [12]. The resonant Auger peak width approaches and merges the normal one with increasing photon energy. Eq. (3) shows that the resonant Auger peak width in the first XAS peak region is the
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
final-state width (0.5–0.6 eV), whereas if the effect of the energydependent unrenormalized relaxation rate is negligible, Eq. (13) shows that the resonant Auger peak width in the second XAS peak region is the sum of the lifetime width of the stable resonantly excited core-hole state and the final-state width, i.e., the sum of the spectator decay width, the participant decay width, and the final-state width. The core-hole lifetime of the stable resonantly excited core-hole state is approximately the same as that of the ionized core-hole state. Note that under this assumption the relaxation rate is determined. Fig. 5 in Ref. [12] shows that the resonant Auger peak width in the second XAS peak region is approximately equal to the normal one measured at far above the threshold. The latter width is the sum of the core-hole lifetime width and the final-state width. The resonant Auger peak width increases linearly from the final-state width to the sum of the core-hole lifetime width and the final-state width. This is because in the transition region from the first XAS peak to the second XAS peak the resonant Auger decay spectrum is a superposition of that of the stable excited core-hole state and that of the metastable one which relaxes to the stable one before the core-hole decays. The latter spectrum arises because of a large photon width. The presence of the latter spectrum affects the variation with photon energy of the resonant Auger peak KE in the same region (see later). 4.2. Normal Auger peak width The normal Auger peak width (0.9 eV [12]) at 33 eV above the ionization threshold is the sum of the core-hole lifetime width (0.38 eV [12,16]) and the final-state width. The final-state width is then 0.52 eV. It agrees well with the resonant Auger peak width (0.5 eV) in the first XAS peak region, which is the finalstate width. Eq. (9) shows that the normal Auger peak width in the second XAS peak region is the sum of the core-hole lifetime width (0.38 eV), the final-state width (0.5 eV), and the relaxation rate. In Fig. 3 we plot the variation with photon energy of the difference between the normal Auger peak width at 1873 eV and the one in the second XAS peak region. The variation with photon energy of the width difference agrees qualitatively with that of the relaxation rate. The deviations are within the experimental accuracy of the intensity ratio of the resonant Auger peak to the normal one limited by a large photon width (Fig. 4(a) in Ref. [12]). As the photon width is not small, the normal Auger decay in the first XAS peak region is mostly that of the core-hole state in which the electron excited to the bottom of the conduction band is delocalized. The normal Auger peak width then becomes the same as that at 1839.5 eV. The present scenario can describe the variation with photon energy of the normal Auger peak width. 5. RAES peak KE 5.1. Resonant Auger peak KE The variation with photon energy of the resonant or normal Auger peak KE is shown in Fig. 6 in [12]. The resonant Auger peak KE shows no dispersion with photon energy except when
19
excited by the highest energy photons. The resonant Auger peak KE shows a constant KE in contrast to a positive linear dispersion with photon energy shown by most other resonant Auger decays. Eq. (13) shows that the resonant Auger peak KE in the second XAS peak region is independent of photon energy and is the same as that at the first XAS peak (1839 eV). In the first XAS peak region the resonant Auger peak KE should show a positive linear dispersion with photon energy. However, it shows almost no dispersion. This is because of the resonant Auger decay of the metastable excited core-hole state which relaxes to the stable one before the core-hole decays. Because of a large photon width the presence of the resonant Auger decay of the metastable core-hole state increases with photon energy. Thus, with an increase in photon energy a resonant Auger peak of constant KE becomes more dominant than that of a positive linear dispersion. This results also in an increase in the resonant Auger peak width and intensity in the same photon energy region. The present scenario explains why the resonant Auger peak KE does not show any dispersion with photon energy. The variation with photon energy of the resonant Auger peak KE shows that at 1843 eV, i.e., 2 eV above the second XAS peak, the KE increases by about 2 eV compared to that at the second XAS peak (1841.1 eV) [12]. According to the scenario by Jackson et al. [12], the resonant Auger peak KE at 1843 eV should increase by 4.2 eV (1843–1838.8 eV) compared to that at 1838.8 eV because the resonant Auger peak KE is expected to follow a positive linear dispersion. The relative resonant Auger peak intensity in the RAES spectrum at 1843 eV is very small and the normal Auger peak intensity dominates (Fig. 4(a) in Ref. [12]). Above the second XAS peak energy (1841.1 eV) where the relaxation rate diminishes, there should be another mechanism by which the resonant Auger peak appears. The KE difference between the resonant Auger peak and the normal one is 2Uc a − Uca. Here Uca is the screened Coulomb hole–particle interaction energy in the stable excited core-hole state, i.e., the one between the 1s hole and the excited electron in the localized state, while Uc a is the one in the final state of the spectator decay of the stable excited core-hole state, i.e., the one between the single 2p hole and the excited electron in the localized state. The normal Auger peak KE is 1607 eV [12], whereas the resonant one is 1611 eV [12] (2Uc a − Uca = 4 eV). Thus, the potential in the two 2p hole final state is much more attractive than that in the single 1s hole state. At 1843 eV the normal Auger decay spectrum dominates in the RAES spectrum. Thus, the excited electron in the conduction band is most likely to delocalize before the core-hole decays. As the relaxation rate diminishes above the second XAS peak (Fig. 3), there is still a possibility that the excited electron remains in the conduction band during the core-hole decay. The excited electron will then be exposed to two holes created by the core-hole decay. As the two-hole potential is much more attractive than the single-hole one, the excited electron is pulled down to the maximum of the conduction band. The excited electron is then further pulled down to the localized state in the delta layer (Fig. 4(a)). Above the second XAS peak the resonant Auger
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M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
Fig. 4. (a) The electron excited at 1843 eV drops to the maximum of the conduction band by a change from a single core-hole potential to a more attractive twohole potential created by the Auger transition of the ionized state. The excited electron then drops to a state localized in the delta layer. (b) The PCI between the photoelectron and the Auger electron emitted. The photoelectron will lose its KE by a change from a single core-hole potential to a more attractive two-hole potential created by the normal Auger decay of the core-hole state. At the same time the Auger electron emitted gains the KE by the screening of one of the two holes by the photoelectron.
peak KE then becomes εA = ω + ε f + ω a − ω b
(16)
Here εf is the 2h1p final-state energy and ωb is the second XAS peak energy (1841.1 eV), while ωa is the first one (1838.8 eV). Eq. (16) shows that above the second XAS line peak the resonant Auger KE shows a positive linear dispersion with photon energy (Fig. 6 in Ref. [12]). The present scenario can explain well the variation with photon energy of the resonant Auger peak KE. 5.2. Normal Auger peak KE The normal Auger peak KE decreases from the first XAS peak region to the second one. Because of a large photon width the excitation to the bottom of the conduction band becomes possible. If the radiationless Raman scattering plays an important role, the normal Auger peak KE in the threshold region should increase and approach the one at far above the threshold with photon energy. However, the variation with photon energy of the normal Auger peak KE is opposite. This indicates the presence of the PCI effect [12]. The excited electron acts as a slow photoelectron. The slow photoelectron recedes from the ionic atomic site experiencing an attractive Coulomb core-hole potential. A fast Auger electron is then emitted as the core-hole decays. If the lifetime of the core-hole is sufficiently short, the fast Auger electron can overtake the photoelectron, which is then exposed to a doubly charged ion core. As the doubly charged ion core potential becomes more attractive than the core-hole potential, the photoelectron will be retarded losing energy, whereas the Auger electron now exposed to a single charged ion core screened by the slow photoelectron, gains energy. The PCI is caused by a sudden change of the Coulomb field that a fast Auger electron experiences on overtaking an initially ejected slow photoelectron [18,19]. When the PCI effect is considered, the normal Auger decay spectrum becomes (Fig. 4(b)): d2 σ |P˜ εc |2 ρ(ε)|G(ε − ω)Vaf (εA )ε|˜ε|2 = dω dεA f
× δ(˜ε − εf − ω + εA ) dε d˜ε
(17)
Eq. (17) becomes d2 σ |P˜ εc |2 ρ(ε)Ac (ε − ω) = dω dεA f
f
×
π|Va (εA )|2 |ε|ω + εf − εA |2 dε Γ +∆
(18)
The normal Auger decay spectrum is modulated by |ε|˜ε|2 where | is the photoelectron wavefunction in the singly charged ion core potential, while |˜ε is the one in the doubly charged ion core potential. When the normal Auger decay intensity measured as a function of photon energy is integrated over the Auger-electron KE of a selected decay channel and summed over the final states of a selected decay channel, the spectrum is a partial XAS spectrum modulated by the energy-dependent branching ratio of the Auger-transition rate of a selected decay channel. 6. Conclusion We discussed the RAES spectra of Si delta dopped layers in GaAs with very thin capping layers using a many-body theory. The resonant Auger peak KE shows no dispersion with photon energy, except when excited by the highest energy photons, in contrast to a positive dispersion with photon energy shown by most other resonant Auger decays. This is explained in terms of the relaxation of the metastable excited core-hole state to the stable one before the core-hole decay. The electron excited to the conduction band either delocalizes leaving the ionized Si to decay by a normal Auger decay or transits to a state localized in the delta layer before the core-hole decays. This results in the presence of both normal Auger decay and resonant one, when the photon energy exceeds the one for the excitation to the bottom of the conduction band. The resonant Auger peak KE is the same as that at the resonant excitation to the empty localized state in the delta layer. The resonant Auger peak width is the sum of the corehole lifetime of the stable excitated core-hole state (the excited electron is in the localized state in the delta layer) and the finalstate width. Thus, the resonant Auger peak width becomes larger
M. Ohno / Journal of Electron Spectroscopy and Related Phenomena 153 (2006) 13–21
than the one at the resonant excitation to the stable excited corehole state. The former width is approximately the same as the normal Auger peak width measured at far above the threshold. The photon energy-dependent relaxation rate is determined from the variation with photon energy of the intensity ratio of the resonant Auger peak to the normal one in the photon energy region in which the electron is excited to the conduction band. The relaxation rate which maximizes at the bottom of the conduction band decreases with photon energy. The variation with photon energy of normal Auger peak width in the photon energy region of the resonant excitation to the conduction band can be explained in terms of that of the relaxation rate. Above the metastable (second) XAS peak the resonant Auger KE shows a positive linear dispersion with photon energy. This is explained in terms of the relaxation of the excited electron in the conduction band to a localized state in the delta layer by an attractive two-hole potential in the final state of core-hole decay. A low resolution of the measurement (a large photon width) explains an increase of the resonant Auger peak width with photon energy and a constant resonant Auger peak KE below the threshold. The present theory is applicable to the case in which a similar relaxation from a metastable excited core-hole state to a stable one on time scale of core-hole decay occurs. A high resolution measurement is desirable. References ˚ [1] T. Aberg, Phys. Scr. T41 (1992) 71. ˚ [2] T. Aberg, B. Crasemann, in: G. Materlik, C.J. Sparks, K. Fischer (Eds.), Resonant Anomalous X-ray Scattering, Elsevier Science, NY, 1994, p. 431.
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[3] G.S. Brown, M.H. Chen, B. Crasemann, G.E. Ice, Phys. Rev. Lett. 45 (1980) 1937. [4] W. Drube, R. Treusch, R. Dahn, M. Greibenow, M. Grehk, G. Materlik, J. Electron Spectrosc. Relat. Phenom. 79 (1996) 223. [5] M. Elango, A. Ausmees, A. Kikas, A. Maiste, E. Nommiset, A. Saar, J. Electron Spectrosc. Relat. Phenom. 72 (1995) 127. [6] W. Drube, R. Treusch, G. Materlik, Phys. Rev. Lett. 74 (1995) 42. [7] O. Bj¨orneholm, A. Nilsson, A. Sandell, B. Hernn¨as, N. M˚artensson, Phys. Rev. Lett. 68 (1992) 1892. [8] W. Wurth, P. Feulner, D. Menzel, Phys. Scr. T 41 (1992) 213. [9] M. Ohno, Phys. Rev. B 50 (1994) 2566. [10] O. Karis, A. Nilsson, M. Weinelt, T. Wiell, C. Puglia, N. Wassdahl, N. M˚artensson, M. Samant, J. St¨ohr, Phys. Rev. Lett. 76 (1996) 1380. [11] C. Keller, M. Stichler, G. Comelli, F. Esch, S. Lizzit, D. Menzel, W. Wurth, Phys. Rev. B 57 (1998) 11951. [12] M.D. Jackson, J.M.C. Thornton, D. Lewis, A. Robinson, M. Fahy, A. Aviary, P. Weightman, Phys. Rev. B 71 (2005) 075313, and references therein. [13] H.W. Haak, G.A. Sawatzky, T.D. Thomas, Phys. Rev. Lett. 41 (1978) 1825. [14] M. Ohno, G. Wendin, J. Phys. B 12 (1979) 1305. [15] W. Drube, A. Lessmann, G. Materlik, Jpn. J. Appl. Phys. 32 (Suppl. 32-2) (1993) 173. [16] P.S. Fowles, J.A. Evans, P.M. Lee, A.D. Laine, P. Weightman, Phys. Rev. B 48 (1993) 14142. [17] S.T. Perkins, D.E. Cullen, M.H. Chen, J.H. Hubbell, L. Rathkopf, J. Scofield, Lawrence Livemore National Laboratory, Evaluated Atomic Data Library, UCRL-50400, vol. 30, 1991. [18] A. Niehaus, J. Phys. B 10 (1977) 1845. [19] M.Yu. Kuchiev, S.A. Sheinerman, Sov. Phys. JETP 63 (1986) 986.