Many-electron singularity in X-ray photoemission and X-ray line spectra from metals

It is pointed out that the spectra of x-ray induced fast photoelectrons from metal should have a characteristic skew line shape resulting from Kondo-like manyelectron interactions of the metallic conduction electrons with the accompanying deep hole in the final state. The same line shape should also occur for the discrete line spectra of x-rays emitted from metals. This mechanism could account for the well known asymmetries observed for Ka lines. 1. Discussion A couple of years ago, Mahan (1967) suggested that an electron excited up to the Fermi surface of a metal from a core state by the absorption of an x-ray (or the converse emission process) would scatter in a singular way, analogous to that occurring in the Kondo effect, from the transient, screened Coulomb, potential of the accompanying deep (and immobile) hole, and thus lead to a singular peak in the soft x-ray absorption (emission) cross section at threshold. More recently, Nozibres and de Dominicis (1969) (to be referred to as ND) have shown that what actually happens is a combination of the above singular electronhole scattering together with a transient and singular re-adjustment of the ground state of the entire Fermi gas to the presence of the effective potential of the hole. Depending on the sign and strength of this potential, these two effects may reinforce or cancel, leading to the possibility of either threshold peaks or zeros in the x-ray absorption cross section. In this paper we discuss two related measurements which, in principle, provide a direct way of studying the transient re-adjustment of the ground state of the Fermi gas to the sudden appearance of the hole potential. These are: (i) the measurement of the detailed line shape of fast photoelectrons emitted as a result of the absorption of monochromatic x-rays by the metal; (ii) the measurement of the line shape of discrete x-ray lines emitted as a result of electron transitions between inner shell states of an excited atom in a metal. In the first case the energy 6,' is a direct measurement of the energy of the hole state Eh (measured relative to the Fermi level) left behind in the metal : E k = W + E , E h w where w is the x-ray energy (h = l), E, the initial ground state energy of the metal, and W is the work function. Thus the maximum photoelectron energy E, corresponds to the ground state of the hole + metal, while photoelectrons emitted below the maximum correspond to events in which the hole + Fermi sea is left in an excited state. Excited states with energies very close (a fraction of an electron-volt) to the ground state are those in which the Fermi sea is excited by the creation of low energy conduction electron-hole pairs (i.e. charge density fluctuations). It turns out that since the energy of creating pairs goes continuously to zero as the momentum transfer to the pair, an infra-red catastrophe occurs in which it is very favourable to produce a large number of very low energy pairs. Thus the photoemission cross section do/dc is changed from a 8-function (for &function I' Now at Department of Applied Physics, Stanford University, Stanford, California 94305, U.S.A. $ On leave of absence from Institut Rudjer BoSkoviC, Zagreb, Yugoslavia. 285 286 S. Doniach and M . sunji.C ingoing x-ray spectrum) in the absence of pair formation to a singular (though integrable) curve tailing off on the low energy side of cmax (figure 1, curve A). In the second type of measurement, a deep hole which has previously been formed by the absorption of an x-ray (or by knocking out an inner shell electron with a fast primary electron) captures an electron from an adjacent inner shell of the atom with emission of an x-ray, i.e. the hole moves ‘up’ to a higher atomic level. When this happens the effective screened potential acting on the conduction electrons changes somewhat, so that again the Fermi sea has to undergo a drastic re-adjustment to the new potential. The emitted x-ray can thus leave the Fermi sea in a many-pair excited state, with accompanying distortion of the x-ray line shape. J I I I 1 1 I I I I -4 -3 -2 I 0 I 2 3 4 Figure 1. Singular line shape for singularity index ci = 0.3: curve A, in the absence of lifetime broadening-broken curve, arbitrary units (equation (9)); curve B, with finite hole lifetime-full curve (equation (18)); energy measured in units of y. In practice both types of event will be rounded out by lifetime effects (figure 1, curve B). However, in the photoemission case, the resulting tailing of the line shape has to be distinguished from events in which the escaping electron loses energy to the Fermi sea. We can, conceptually, distinguish two types of such energy loss : energy loss to the bulk metal (e.g. plasmon emission), and energy loss to surface states of the metal (e.g. surface plasmons). In the analysis which follows we assume that the probability of bulk energy loss by the escaping photoelectron is a function of thickness of the metallic target, and goes to zero as the film thickness goes to zero. This will not apply to surface energy loss, but presumably the probability of this process will decrease with increasing escape energy of the photoelectron, so it can be distinguished by varying the incident photon energy. In contrast, the excitations of the Fermi sea accompanying the final hole state are an intrinsic property of the metal, so they should not depend either on film thickness (above a few atomic layers) or on incident photon energy. 2. Photoemission In the photoemission case the final state of the system may be written I yf) = cl 1 yhole> (1) where c l is a creation operator for the fast photoelectron and (Yhole) is a wave function Many-electron singularity in x-ray photoemission 287 of the hole + Fermi sea. The photoemission scattering cross section may then be written, to lowest order in the x-ray field, as states where I Yg) is the initial ground state of the metal, j is the electron-hole current coupling to the x-ray field : j = C j k ' ( b + C : + bck,) (3) k' (b+ is the hole creation operator) and 6h = E , E , + W We shall treat the current matrix elements j , as constant. Introducing the Hamiltonian of the hole + Fermi sea (ND) = Hcond + we now make the important approximation, discussed in $1, of neglecting all interactions of the escaping electron with the other electrons in the metal. We may thus write da 1 " cc ReGJo dt(b(t)b+(O)) exp {i(o 6k)t) d6k where b+(t) = exp (iHt)b* exp (iHt). The hole-hole correlation function in ( 5 ) is directly related to the hole propagator introduced by ND. By noticing (ND, Langreth 1969) that, owing to the large energy of formation of a hole, only one hole can be present at a time, this correlation function may also be written directly in terms of the Hamiltonian Hc& of the conduction electrons in the absence of the hole, and the Hamiltonian Hcond + H, of the electrons in the presence of a static hole potential :