Resonance Interatomic Auger Transitions

Abstract

The interatomic Auger transitions in compounds containing atomic components with core levels close in energy are studied theoretically and experimentally. The Coulomb transitions of holes between such levels lead to resonant enhancement of the Auger spectra (with respect to the energy difference between the levels). Interatomic Auger transitions involving higher levels are formed by shaking up electrons due to the dynamic field of photo-holes produced in the process of absorbing X-ray radiation and during the transition. These effects are observed experimentally in the XPS and Auger spectra of CuInSe₂-type materials.

INTRODUCTION

The investigation of interatomic Auger transitions is of significant importance for studying the evolution of excited stated in substances. It is of special importance to understand the Auger transitions in photoelectric transducers, because these transitions play a determining role in energy losses and limit the efficiency of transducer operation [1]. It was shown in [2, 3] that photoemission associated with the ground electronic level of a given atom “А” can be significantly altered in terms of its intensity when the photon energy overcomes the absorption edge of the core level of a neighboring atom “B”. This effect is called multiatom resonant photoemission (MARPE). Its macroscopic description can be given in terms of a resonance optical dielectric model that considers a change in the complex permittivity in the process of passing the corresponding resonance [4–6].

In this work, Auger processes with no resonance photons are considered theoretically and experimentally. A photohole decays by means of a nonresonant Auger transition with the emission of an electron by a neighboring atom. It is shown that this process is significantly enhanced if the core levels of neighboring elements have nearly identical energies. From this point of view, it is possible to speak about a resonant interaction between the core levels involved in the Auger transition.

AUGER PROCESS IN A COMPOUND WITH TWO LEVELS CLOSE IN ENERGY

Let us consider the scheme (Fig. 1) of the interatomic Auger transition by the example of the СuInSe₂ compound. A Cu 2р hole that appears as a result of photoionization (the binding energy is 933 eV) due to the Coulomb interaction is filled by an electron from the upper Cu 3р-state (75 eV); the energy release in this process is spent on the emission of an electron of the neighboring atom of In 4d (17 eV) to the free state f. A detector measures the intensity of the emission of such electrons and their kinetic energy or the СuL₃M_2,3, InN_4,5 Auger spectra. Usually, the intensity of the interatomic transition is vanishingly small as compared with that of the intra-atomic transition, for example, Сu (L₃M_2,3V) (V designates the valence state), since the distances determining the Coulomb interaction energy ${{{{e}^{2}}} \mathord{\left/ {\vphantom {{{{e}^{2}}} {\left| {{{{\mathbf{r}}}_{1}} - {{{\mathbf{r}}}_{2}}} \right|}}} \right. \kern-0em} {\left| {{{{\mathbf{r}}}_{1}} - {{{\mathbf{r}}}_{2}}} \right|}}$ between electrons inside an atom is significantly smaller than the interatomic distance.

Fig. 1.

Diagram of the multiatomic Auger processes Cu 2p Cu 3p In 4d (a) and Cu 2p In 4p In 4d (b) in СuInSe₂ taking into account the interaction between the Cu 3p and In 4p core levels.

Let us show that the intensity of the interatomic Auger transition increases significantly, if the neighboring atom has a core level close in energy to the level of the central atom. In our example, it is the In 4p level with a binding energy of 73.5 eV, which is shifted by 1.5 eV only relative to the energy of the Cu 3p level. The probability of virtual hole transitions between levels increases (Fig. 1a); the real (in the final state) transition of a hole from one atom to another (Fig. 1b) also becomes possible. The rate of the transition from the initial state i to the final state j (generally speaking, decaying) is defined by the quantity

$${{I}_{{ji}}} = \frac{1}{{\hbar {\pi }}}\operatorname{Im} {{G}_{{jj}}}({{E}_{i}}){{\left| {{{A}_{{ji}}}({{E}_{i}})} \right|}^{2}}.$$

(1)

Here,

$$\begin{gathered} {{A}_{{ji}}}(E) = {{\left( {V + VG(E)V} \right)}_{{ji}}}, \\ G(E) = {{\left( {E - H - V - i{\gamma }} \right)}^{{ - 1}}} \\ \end{gathered} $$

(2)

is the total Green’s function with the Coulomb interaction V leading to Auger transitions between the eigenstates i and j of the zero Hamiltonian H. The imaginary component in equation (1) reflects the energy conservation law (at γ → 0) with consideration of the state decay (at a finite γ).

In the first order, the amplitude of the Auger transition is equal to the matrix element of the Coulomb interaction between the initial and final states:

$${{A}_{{ji}}} = {{V}_{{ji}}} = \left\langle {1,3\left| V \right|c,f} \right\rangle .$$

(3)

On the diagram (Fig. 1), the state с is a hole at the Cu 2p level; f is the Auger electron recorded by the detector; state 1 is the Cu 3p hole; state 3 is a hole at the In 4d-level. The amplitude (diagram 1a) of the Auger transition with consideration of the interaction between states 1 and 2 (In 4p) can be written as follows:

$$\begin{gathered} {{A}_{{ji}}} = \frac{{\left\langle {1,1{\kern 1pt} '\left| V \right|2,1{\kern 1pt} '} \right\rangle }}{{E - {{E}_{2}} - {{E}_{3}} - i{\gamma }}} \\ \times \,\,\frac{{\left\langle {2,2{\kern 1pt} '\left| V \right|1,2{\kern 1pt} '} \right\rangle }}{{E - {{E}_{1}} - {{E}_{3}} - i{\gamma }}}\left\langle {1,3\left| V \right|c,f} \right\rangle . \\ \end{gathered} $$

(4)

In what follows, we will denote the matrix element of the Coulomb electron transfer between atomic levels 1 and 2 (the numerator in (4)) by symbol W; the matrix element (3), by symbol V. Calculating the first terms of the series with respect to interaction, we obtain the intensities of the Auger transitions in different orders of scattering theory.

The Auger-line intensity as a function of the electron kinetic energy and its integrated intensity (the line power) in the lowest order are given by the following equations:

$${{I}_{0}}(e) = \frac{{{{V}^{2}}}}{{\pi }}\frac{{\gamma }}{{{{e}^{2}} + {{{\gamma }}^{2}}}};\,\,\,\,{{S}_{0}} = \int {de{{I}_{0}}} = {{V}^{2}}.$$

(5)

The quantity $e = {{E}_{{{\text{kin}}}}} - ({{E}_{1}} + {{E}_{3}} - {{E}_{c}})$ shows the deviation of the kinetic energy of the Auger electron from its nominal value.

The rise of the hole from level 1 to the level of the neighboring atom 2 (Fig. 1b) reduces the energy of the atom remnant by the value ${{E}_{{21}}} = {{E}_{2}} - {{E}_{1}}$ and generates a spectral line of the following form:

$$\begin{gathered} {{I}_{1}}(e;{{E}_{{21}}}) = {{W}^{2}}{{V}^{2}}\frac{1}{{\pi }}\frac{1}{{{{e}^{2}} + {{{\gamma }}^{2}}}}\frac{{\gamma }}{{{{{\left( {e - {{E}_{{21}}}} \right)}}^{2}} + {{{\gamma }}^{2}}}}; \\ {{S}_{1}} = \frac{{2{{W}^{2}}{{V}^{2}}}}{{{{E}_{{21}}}^{2} + 4{{d}^{2}}}}. \\ \end{gathered} $$

(6)

The virtual hole transition between levels 1 ↔ 2 gives the addition:

$$\begin{gathered} {{I}_{2}}(e;{{E}_{{21}}}) = {{W}^{4}}{{V}^{2}}\frac{1}{\pi }\frac{\gamma }{{{{{\left( {{{e}^{2}} + {{\gamma }^{2}}} \right)}}^{2}}}}\frac{1}{{{{{\left( {e - {{E}_{{21}}}} \right)}}^{2}} + {{\gamma }^{2}}}}, \\ {{S}_{2}} = \frac{{{{W}^{4}}{{V}^{2}}}}{{{{E}_{{21}}}^{2} + 4{{\gamma }^{2}}}}\left\{ {\frac{1}{{2{{\gamma }^{2}}}} + \frac{4}{{{{E}_{{21}}}^{2} + 4{{\gamma }^{2}}}}} \right\}. \\ \end{gathered} $$

(7)

The hole transition from level 1 to the neighboring level 2 taking into account the virtual 2 ↔ 1 transition generates the line

$$\begin{gathered} {{I}_{3}}(e;{{E}_{{21}}}) = \frac{{{{W}^{6}}{{V}^{2}}}}{\pi }\frac{1}{{{{{\left( {{{e}^{2}} + {{\gamma }^{2}}} \right)}}^{2}}}}\frac{\gamma }{{{{{\left( {{{{(e - {{E}_{{21}}})}}^{2}} + {{\gamma }^{2}}} \right)}}^{2}}}}, \\ {{S}_{3}} = \frac{{{{W}^{6}}{{V}^{2}}}}{{{{{({{E}_{{21}}}^{2} + 4{{{\gamma }}^{2}})}}^{2}}}}\left( {\frac{1}{{{{{\gamma }}^{2}}}} + \frac{{16}}{{({{E}_{{21}}}^{2} + 4{{{\gamma }}^{2}})}}} \right). \\ \end{gathered} $$

(8)

Figure 2 shows the contributions to the Auger spectra from the above-considered scattering channels (formulas (5)–(8)) at an energy difference between levels equal to their width of E₂₁ = 2γ. The thin solid line shows the intensity in the least order I₀ (5). The dotted lines are graphs of the functions I₁(e), I₂(e), and I₃(e); the thick solid line shows the sum of all four contributions. Consideration for the contribution of neighboring atoms leads to a significant increase in the Auger-emission intensity in the case where E₂₁ = 2γ. It should be noted that we summarize the intensities rather than the amplitudes of the transitions, since intensive scattering results in the chaotization of the phases of wave functions and in the attenuation of interference effects [7].

Fig. 2.

Intensity of Auger emission (the thick solid line) and contributions of channels (5)–(8) at an energy difference equal to the level width E₁₂ = 2γ.

Figure 3 shows the dependence of the power of the S₀ – S₃ processes on the value of the energy difference between levels E₂₁/γ at a fixed value of the matrix element of the Coulomb transfer W = 1.41γ. The use of a finite series of the perturbation theory at zero energy difference between levels is not completely justified; however, even at E₂₁ = 2γ, this series converges quickly and the increase in the Auger-emission power due to the interaction between core levels of neighboring atoms by two-three times is quite possible. The effect of an increase in the Auger-emission power quickly decreases as the energy difference grows and it becomes vanishingly small at E₂₁ > 5γ.

Fig. 3.

Integrated intensity of the Auger line as a function of the energy difference between the levels E₂₁ at W = 1.41γ: power of the S₀–S₃ processes and their sum (the thick solid line).

The presence of a resonant (with a close value of energy) level at the neighboring atom significantly increases the Auger-transition probability. The contributions of the high-energy level deform the line towards increasing kinetic energy. The spectrum becomes narrower due to virtual excitations. It should be taken into account that the final states in real atoms are a level multiplet; therefore, several lines with different energies emerge instead of one line.

ANALYSIS OF EXPERIMENTAL AUGER SPECTRA

The experimental X-ray photoemission spectra (XPS) and Auger spectra of chalcopyrite СuInSe₂ at various excitation energies were obtained at the BESSY II Russian-German laboratory (Berlin). Figure 4 shows the Auger spectra of Cu in СuInSe₂ obtained at a photon energy of 1200 eV. For comparison, the Auger spectrum of metallic copper obtained for the case of MgK_α radiation with an energy of 1253.6 eV [8] is also presented. On both these curves, one can clearly see the CuL₃VV intra-atomic Auger transitions (the maximum of the kinetic energy is 918 eV) and a triple Auger-line formed by the CuL₃M_{2, 3}V transition (the principal maximum is 838 eV, the multiplet splitting as a result of summation of the moments of two holes 3p and 3d). At the energy values located 20 eV above the main lines on the curves shown in Fig. 4, one can note their replicas originating from the CuL₂ hole. The formation of the CuL₃VV Auger line in the related compound Cu (In_0.9Ga_0.1Se₂) in the process of the photon-energy overcoming the excitation edge of the 2р level, the fine line structure, and the energy of the Hubbard repulsion of two holes in the valence band are studied in detail in [9].

Fig. 4.

Intra-atomic СuLMM and interatomic CuLM InN Auger lines in СuInSe₂ (the solid line) and the spectrum of pure copper [7] (the dotted line).

It is seen in Fig. 4 that in СuInSe₂ at an energy of 14 eV below the Cu L₃VV peak, the interatomic CuL₃ InN_4,5V Auger line with a width of 25 eV appears, while this line is absent in the spectrum of pure copper. A similar structure is also seen below the CuL₃M_2,3V lines. This is a result of the second interatomic transition CuL₃M_{2, 3} InN_4,5. Let us note that the CuM_2,3 hole can be completed with the N_2,3 hole whose binding energy is lower only by 1.5 eV. Hence, in the XPS spectra of chalcopyrite СuInSe₂, interatomic Auger transitions are observed. Their intensity is enhanced due to resonant interaction between the Cu 3p and In 4p levels with close energies.

The model developed in the previous section is directly applicable to the interatomic Auger transitions CuL₃M_2,3InN_4,5 and CuL₃InN_2,3InN_4,5. But what is the mechanism of enhancement of the Cu L₃ In N_4,5V transition? According to our opinion, it is highly likely that the scenario is as follows. First, a Cu 2p-hole is filled by a Cu 3p-electron with the emission of an Auger-electron from the valence state; next, the transition of the Cu 3p-hole to the In4p state of the atom of indium occurs; this process proceeds to completion by the intra-atomic electron transition at the indium atom In4d → In4p with the transfer of the energy excess to the Auger electron. As a result, we come from the initial Cu 2p hole state to the final In N_4,5V hole state by means of two intra-atomic transitions and the resonant transfer of the hole between atoms, which have quite high probabilities.

Figure 5 shows a fragment of the photoelectron spectrum of СuInSe₂ in the region of Auger transitions that accompany the decay of the In 3d_{3/2, 5/2} hole doublet (the binding energies are 451.4 and 443.9 eV) obtained at a photon energy of 600 eV, as well as the spectrum of pure indium for the case of MgK_α radiation of 1253.6 eV [8]. Both of these spectra have similar structures, except for the contribution of the direct photoemission from the Se 3s state (370 eV) in СuInSe₂. All the Auger transitions have an intra-atomic character. The most intensive line is the In 3d4d4d doublet with a kinetic energy of 401.5 and 408.5 eV. The line intensities in descending order are as follows: broad lines with two holes in the states 4p4d (with a center at 336 eV), 4s4d (296.5 and 301.5 eV), and 3p3p (267.5 and 278.5 eV). Apparently, a maximum with an energy of 13 eV below the principal peak occurs due to the energy loss spent on plasmon generation. A similar characteristic energy loss of 8.5 eV is also observed in metallic indium.

Fig. 5.

Photoemission spectrum of СuInSe₂ obtained at a photon energy of 600 eV (the solid line) and the spectrum of pure indium at 1253.6 eV [7] (the dotted line).

There is a small peak at an energy of 423 eV from the In 3d4dV Auger transition; the In 3d VV transition does not manifest itself at all. The In 4p hole that appears due to the decay process may move with considerable probability to the copper atom to the Cu 3p state; however, the probability of the transition of the valence electron V to the Cu 3p level is low (significantly lower than the probabilities of transitions V → Cu 2p and In 4d → In 4p initiated by Cu 2p-hole decay). As a result, the mixing of the Cu 3p and In 4p states does not manifest itself in the Auger spectra generated by In 3d-hole decay. Meanwhile, in the process of evolution of the excited state with a hole at Cu 2p, favorable conditions appear for the interatomic Auger processes.

CONCLUSIONS

Theoretical justification and experimental confirmation are given to the idea that the increase in the cross sections of the Auger processes due to interaction between the core levels of neighboring atoms with close energies makes it possible to observe the Auger transitions in the X-ray band. The experimental XPR spectra of compounds based on chalcopyrite CuInSe₂ show the intensive interatomic Auger transitions CuL₃M_2,3 InN_4,5 and CuL₃ InN_4,5. The resonant enhancement of the interatomic Auger-electron emission Cu L₃M_2,3 In N_4,5 is described by the theory of multiple scattering with consideration of the closeness in energy of the Cu 3p and In 4p levels. The sudden occurrence of photo- and Auger holes creates a dynamic field with a broad frequency spectrum, which initiates the shaking up of electrons of neighboring atoms. This process significantly increases the probability of the interatomic Auger transition CuL₃ InN_4,5V. In compounds with a narrow valence band (for example, in those of 3d-type) and a localized core level with a moderate binding energy (In 4d, 17 eV), strong Coulomb interaction between electrons and holes at neighboring atoms appears, which creates favorable conditions for the occurrence of intensive interatomic transitions in the soft X-ray band.