Multiplet effects in Resonant X-ray Emission

After a short discussion of all conventional core level spectroscopies within the single particle model, the effects of the coupling of the core and valence wave function on the x-ray emission spectral shapes is discussed. It will be shown that these so-called multiplet effects strongly affect all x-ray emission spectra taken around the metal 2p resonances. In case of 1s resonances, valence band x-ray emission is not affected, but the spectral shapes of 1s2p and 1s3p x-ray emission can only be sensibly described with the inclusion of multiplets. A special example is the resonant excitation into the pre-edge region, which gives rise to a quadrupole resonance. CORE LEVEL SPECTROSCOPIES All core level spectroscopies will be introduced using a simplified single particle model. This model is not expected to give a correct interpretation of the spectral shapes and intensities observed, but serves as a starting point on which more complete models build upon. The Single Particle Approximations In order to get a first idea of how the x-ray and electron spectra will look like we use a simplified single particle model. A series of approximations have been made: 1. Density-of-States approximation: The assumption that the single particle Density-of-States (DOS), as calculated for example by Local Spin Density (LSD) calculations or by real space Multiple Scattering codes, gives a good account of the electronic structure of the ground state. 2. Spectroscopy approximation: The assumption that the ground state DOS can in fact be used to describe spectroscopic transitions. This implies, for example, that it is assumed that in photoemission the valence hole does not cause a redistribution of the electronic states. 3. Core hole approximation: The assumption that the core hole created in x-ray absorption does not modify the DOS. 4. Matrix Element approximation: The assumption that the transition matrix elements are constant over the energy range analyzed. Some of these omitted effects are relatively easy to include. For example, it is in general straightforward to include the matrix element effects. In electronic structure models based on real-space multiple scattering, it is also straightforward to include the core hole. The closely connected approximations (1) and (2) are more difficult to include and are related to electron-correlation effects that are not included in meanfield electronic structure methods. The Core Hole Spin-Orbit Splitting The coupling of the orbital and the spin-moment is given by the spin-orbit interaction, which is essentially a relativistic effect. The spin-orbit interaction is large for core holes and in general two peaks or structures will be visible in the spectrum, separated by the core hole spin-orbit splitting. The relative intensity of the 2p1/2 and 2p3/2 peak is 1:2 (given by the degeneracy of the states) and that of the 3d3/2 to 3d5/2 peak is 2:3. This rule is general for all core level spectroscopies. The rule breaks down only if another interaction is able to mix the 2p1/2 and 2p3/2 states. It will be discussed below that an important interaction is the overlap of the core hole wave function with the valence state wave functions, or in other words, the coupling of the core and valence moments. This coupling destroys the simple picture sketched above. X-ray Absorption and X-ray Photoemission It has been shown that the x-ray absorption (XAS) process can be described with the dipole approximation, under which the orbital quantum number must be modified by one while the spin quantum number is conserved. This implies that if a 1s-core electron is excited one observes the empty DOS of p-character. This p-projected unoccupied DOS is abbreviated as Cp. Similarly the excitation of a p core electron probes Cs plus Cd. The Fermi level is reached if the x-ray energy !ω is exactly equal to the binding energy of the core hole (E1s). The unoccupied DOS is reproduced by the spectrum with energies C=!ω+E1s, with the 1s binding energy given with a negative number. The same core hole excitation process can excite an electron out of a solid. Its kinetic energy (Ek) can be detected and gives directly the binding energy (Ec) of the core electron as: Ec=Ek-!ω, with the binding energy given as a negative number (and omitting the effects of the work function). The 1s XPS spectral shape thus consists of a single line. If a valence electron (V) is excited, exactly the same formula implies: Ec = Ek !ω. An electron at the Fermi level is detected if the kinetic energy of the emitted electron exactly equals the x-ray energy, i.e. the occupied DOS can be determined by subtracting the x-ray energy from the kinetic energy: V=Ek-!ω, again using negative numbers for the occupied DOS. In principle the dipole selection rule applies again, but because the emitted electron can have any angular momentum in practice the dipole selection rule is dysfunctional and one observes the total occupied DOS (within the imposed approximation on matrix elements). Inverse Photoemission (IPES) is the inverse process of photoemission: an electron is directed towards the sample, it is absorbed in the unoccupied DOS, thereby emitting an x-ray. X-ray Emission and Auger The core hole decay can take place radiatively by a x-ray emission (XES) process or non-radiatively by an Auger process (AES). XES follows the same selection rules as x-ray absorption and a 1s-core hole can be filled by a core 2p electron or a valence p electron. If a 2p core electron fills the 1s core hole the x-ray energy gives the difference in binding energy of the two core states E2p -E1s = !ω'. If a valence electron fills the 1s core hole one maps the p-projected occupied DOS: V= !ω'+E1s. Instead of the dipole matrix element r, decay of a core hole can occur via the electrostatic two-electron integrals , where a, b, c and d are electronic states. An example is the matrix element <1s εd|1/r|2p2p>. In this Auger process one 2p electron fills the 1s core hole and the other 2p electron is excited out of the solid as a free electron. In this 1s2p2p (or KL2,3L2,3) Auger process the final state contains two 2p-core holes. The kinetic energy of the emitted electron equals Ek=E2p+E2p-E1s. If a 2p-core electron and a valence electron take part in the Auger process one can detect again the occupied DOS. Neglecting the Auger matrix elements (selection rules), we assume that the total occupied DOS is detected: V= Ek -E2p+E1s. It is also possible that two valence electrons take part in the Auger process. In this 1sVV Auger process one detects the self-convolution of the occupied DOS: 2V= Ek +E1s. Resonant PES, AES and IPES In the foregoing, core hole creation (XAS) has been described separately from core hole decay (XES and AES). However as soon as core hole creation takes place decay occurs. This implies that close to the XAS absorption resonance's, the decay processes can be different from off-resonance excitations. The only effect within the single particle model is that the excited electron can take part in the decay process. This creates additional decay channels not present in off resonant or normal AES and XES. The best-known resonant spectroscopy is resonant photoemission (R-PES). Within the single particle model this can be described as the two-step process of x-ray absorption followed by Auger, i.e. Φ0→→ 1sC →→ V, where Φ0 is the ground state and 1s a 1s core hole. In the final state, a hole exists in the valence band and one measures the occupied DOS as in a normal photoemission process, but a difference is caused by the matrix element. The direct PES channel and the indirect XAS+AES channel have the same initial and final states, hence they interfere with each other. There is another possible R-PES channel, i.e. Φ0→→ 1sC →→ V+V+C. In this case the C electron of the intermediate state does not participate in the Auger decay and this process is called the spectator channel. The process in which the C does participate is called the participator channel. The final state of the spectator channel has two valence holes (V) plus an extra electron in C. This can be viewed as a normal photoemission final state, plus a V to C excitation. This final state cannot be reached by normal PES within the single particle model. It plays a crucial part of all many body descriptions of core level spectroscopy, i.e. it is essentially the shake-up channel. Going to off-resonance conditions, the spectator channel disappears, as the C electron becomes a free electron. Another resonant photoemission process is for example 2p3pV R-PES, in which first a 2p-core hole is created at resonance that subsequently decays by Auger to a 3pcore hole, i.e. Φ0→→ 2pC →→ 3p, 3p+V+C. The final state in the participator channel is again equal to normal 3p excitation, while the spectator channel adds a V to C excitation. Resonant Auger spectroscopy (R-AES) is the same as RPES. The name R-AES will be reserved however for those processes that cannot be reached by direct photoemission. In practice this are all final states with two core holes present in the final states. An example is the 2p3p3p R-AES process, in which first a 2p-core hole is created at resonance that subsequently decays by Auger to two 3p-core holes, i.e. Φ0→→ 2pC →→ 3p3pC. Only spectator R-AES processes exist, because if the C electron participates one never creates a two-hole final state. If one moves away from resonance, the created C electron becomes a free electron, implying that the R-AES channels disappear. The reverse process, resonant inverse photoemission (R-IPES), is also possible. If one excites with electrons that have a kinetic energy equal to the binding energy of a core state, an Auger process can occur in which both the core electron and the impinging free electron are transferred to an electron in the conduction band. In a second step, a conduction electron can decay by XES to the core hole. This could again be called a participator channel. In the spectator channel a valence electron decays to the core hole, i.e. Φ0→→ 2pCC →→ C, C+C+V.