Physica Scripta. Vol. T41, 59-66, 1992. Correlation Effects in Auger CVV Spectra from Partially Filled Bands Michele Cini Dipartimento di Fisica, Universita’ di Roma 11, Via 0. Raimondo, 1-00173 Roma, Italy Received September 3, 1991 : accepted September 6, 1991 Abstract Auger Line Shape Analysis from solids with open bands has made con- siderable progress in recent years. Calculations on Graphite and Pd were encouraging, and new developments in the formalism look promising. However, it must be recognized that the current theory remains much more phenomenological and less predictive and reliable than it is for closed bands. In some sense, the comparison with experiments is more successful than our real understanding deserves. This is hardly surprising, in view of the much more fundamental character of the problems involved. An attempt will be made to assess the present state of the theory and fore- shadow the directions of future research. 1. Introduction When Powell [I] made the experimental discovery of the existence of band-like and atomic-like CVV spectra, showing that the line shapes of AI and Ag are drastically different, it was the Ag spectrum that appeared “anom- alous”, since that of A1 was qualitatively consistent with Lander’s theory [2]. It is well known by now that the quasi- atomic peaks are two-hole resonances that arise in closed bands when U/W (correlation energy to band width ratio) exceeds a critical value. On the other hand, correlation effects are not too evident in the CVV spectra of simple metals, that are band-like and rather featureless [3, 41; per- turbation theory methods suitable for such spectra have been developed [5, 61 and applied to Li, Be, Mg and Al. Here we shall concentrate on narrow open bands, where the potential energy due to the hole-hole repulsion can be com- parable to the kinetic energy; the simple U/W criterion must be modified to allow for many-body intra-band excita- tions, producing polarization and shake-up effects. The theory Auger CVV line shapes is much more involved when the valence band is partially filled. In order to fully appreciate the problem, we must follow the reason- ing which underlies the closed-band theory [7, 81 which is based on the explicit assumption of fully occupied orbitals. Note that this assumption does not only exclude conduction bands, but implies that most semiconductors and insulators also need to be treated by open-band theory. It is fairly obvious that it reduces the Auger final state to a two hole state, that we can deal with by exactly solvable models; but there is more, because the limitation enters at various stages. The closed-orbitals assumption explicitly enters when we define the local two-hole states at the site where the Auger decay occurs (site 0 in the lattice, say) I m1m2 a> = Ck2, 0 Cil, + I vac>, (1) where CA,, creates a hole of magnetic quantum number m and spin a and I vac) is the hole vacuum. For closed bands, it is easier to justify the final state model Hamiltonian; this is taken to be H = H, + H,, where H, is any one-body operator deemed appropriate, H, is the hole-hole repulsion at site 0, namely and UmlmZmjm4 are Coulomb integrals. Actually, these will be “screened” by the solid, but we may try to ignore the detailed dynamics as long as the valence band is not directly responsible for that. Moreover, no term in this Hamiltonian depends on core states, and this is quite reasonable for closed bands, that cannot polarize in the presence of the core hole. For the same reason, we may drop the core degrees of freedom when we write down the Fermi golden rule expression, and then the transition probability depends on the two-holes density of states matrix D,,,,, mjm4, a(co), that is the Fourier transform of the correlation function Dmimz, m3m4, At) = (m3 m4 a I e-iHr I m1m2 0); (3) its independent-particle counterpart D~lm2, m3m4(t) may be obtained in principle from H, . By a Laplace transformation we obtain a matrix D,(s) whose real part for s = Of - iw is TL times the density of states matrix we are looking for. The exact solution is obtained by solving the equation D,(s) = D:(s) - iD,(s)W,D:(s), Wmimzmjmq, - - Umimtmqmj Wmimzmjmq, + - Umimzmqmj - Umimzmjmq. (4) (5) (6) where the interaction matrix W has elements - - The Wannier functions can be expanded in local orbitals, and since the interatomic transitions are negligible it is natural to replace the Auger matrix elements by their atomic values, that may often be obtained from the liter- ature, provided the atomic configurations in the solid and in the free atom are not too different. If the orbitals are filled in both cases, this is more acceptable. In several cases this approximation turns out to be adequate, but this is no general conclusion [SI, since the potential seen by the Auger electron may be modified by the environment. This theory explained the quasiatomic nature of the spectra of several d-band metals, like Ag, Cd, Zn [9], Cu [lo], In [11] and also Ni [12], despite the partial occu- pancy of the band in the latter case (9.4d electrons per atom). Sawatzky [13] argued that the Auger decay occurs in a locally filled d’’ situation induced by relaxation around the primary core-hole; below, we shall find that even for almost completely filled bands the quasi-atomic peaks exist, with some shift and broadening with respect to the filled band limit, and a broadening is evident in the experimental Physica Scripta T41