Electron spectroscopy of disordered metal alloys

Journal of Electron Spectroscopy and Related Phenomena 178–179 (2010) 100–111

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Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec

Review

Electron spectroscopy of disordered metal alloys P. Weightman a,∗ , R.J. Cole b a b

Physics Department, University of Liverpool, Oxford Street, Liverpool L69 3BX, UK SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK

a r t i c l e

i n f o

Article history: Available online 14 May 2009 Keywords: XPS Auger AES

a b s t r a c t We review the contribution that electron spectroscopy has made to the understanding of the electronic structure of disordered metal alloys. In this review we consider the results of direct studies of the conduction band densities of states using photoelectron techniques and indirect studies of the correlation energies, charge transfers and electron screening through indirect probes combining photoelectron and Auger spectroscopies. It is clear that results obtained from a variety of approaches have made major contributions to the development of both empirical models and first principles treatment of alloy electronic structure. © 2009 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4. 5.

6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction band densities of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron correlation: the Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron screening, charge transfer and s–d hybridisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Auger parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Auger profiles and electron screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CuPd and AgPd alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction It has been difficult to develop a first principles theory of the physical and electronic structure of disordered alloys [1]. For ordered systems the periodic potential makes it possible to solve Schroedingers equation by a variety of methods and if necessary the lattice constants of such systems can be easily determined by X-ray diffraction to provide input into the calculations [2]. However when the system is disordered it is not clear that the physical structure is periodic on a local scale even when it is possible to determine the compositional dependence of average lattice constants using X-ray diffraction. Indeed the use of the extended X-ray absorption fine structure (EXAFS) technique has shown that in some systems at least there are local lattice relaxations that are not revealed by X-ray diffraction results [3–5]. Such local lattice relaxations have

∗ Corresponding author. E-mail address: [email protected] (P. Weightman). 0368-2048/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2009.05.003

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been shown to have a significant influence on the local electronic structure of disordered alloys and pose a significant challenge to first principles theories such as the coherent potential approximation [6] that rely on a periodic potential in order to calculate the electronic structure. Electron spectroscopy provides a number of tests of the results of theories of the electronic structure of disordered alloys. The valence band density of states can be probed directly by photoemission and although these are excited state measurements there are adequate theoretical techniques for relating quasi-particle energies to the ground state density of states [7]. The contribution of individual elements to the total density of states can be probed by photoemission at the Cooper minimum [8–22] and Auger spectroscopy [23–29] and this latter technique can also reveal the on-site electron correlation energy [24,26]. X-ray photoelectron and Auger spectroscopies show shifts in core-level binding energies and Auger kinetic energies that depend on alloy composition and the analysis of which, when combined in “Auger parameters” has shown the need to consider contributions from the initial state charge distribution and the

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contribution from electron screening of the final state core-holes [30–39]. More recently the observation of disorder broadening of corelevel photoelectron line shapes in high resolution photoelectron spectroscopy provides insight into the local atomic environment [40] and the dependence of the atomic core potential on the orbital character of the initial and final state charge distribution, the number and nature of nearest neighbours and local lattice relaxation [41]. This latter subject has had a major impact on our understanding of the electronic structure of disordered alloys and is reviewed in a separate article [42]. While the comparison of experimental results obtained from electron spectroscopy with the results of first principles theory have often lead to controversy it is becoming clear that the challenge presented by electron spectroscopy has been a significant driving force in improving theoretical treatments of disordered alloys and this together with advances in computational techniques is giving rise to an improved understanding of this important class of materials. In this short review we provide a brief history of the development of the field. We have omitted a detailed discussion of surface alloys and alloy surfaces since this subject has been extensively reviewed by a series of articles in a recent book [43] and in a recent review article [44]. The focus will be on bulk properties though in assessing the results reported in the literature it is necessary to be aware of the possible influence of surface effects arising from the low escape depths of photoelectrons. 2. Experimental considerations The experimental techniques of electron spectroscopy have been reviewed many times [45–47] and will not be considered here. The principle techniques for investigating the electronic structure of disordered alloys are X-ray emission spectroscopy (XES), photoelectron and Auger electron spectroscopies excited by fixed energy photon sources such as X-ray anodes (XPS) or He discharge lamps (UPS) or continuously variable synchrotron radiation and, for probing empty states, Bremsstralung isochromat spectroscopy (BIS) and X-ray absorption spectroscopy (XAS). The electronic structure of disordered alloys is of course closely related to their physical structure and the latter can be probed by X-ray diffraction and EXAFS. A crucial issue in the study of disordered alloys is the preparation of clean surfaces. Surface alloys are often prepared by elemental deposition onto clean surfaces in vacuum [43] and these processes can be carefully controlled. Good quality single crystals of disordered alloys are rare and even when they are available it is difficult to devise methods of preparing clean surfaces that do not result in the surface enrichment of one alloy component. This is also a problem in preparing clean surfaces of polycrystalline specimens since the common techniques of ion bombardment and annealing are each known to lead to surface segregation in some cases [48]. The best method of avoiding surface segregation is by mechanically scrapping in vacuum but in all cases it is necessary to consider whether the surface composition is representative of the bulk. 3. Conduction band densities of states The most direct method of investigating the electronic structure of disordered alloys is by measuring the binding energy distribution of the conduction band density of states by photoelectron spectroscopy. For single crystals of disordered alloys it is possible to use high resolution UV photoelectron spectroscopy to map the energy verse momentum dependence of the occupied density of states. The shortage of good single crystals means that there are few studies of this kind [49–51]. In contrast the literature on studies of polycrystalline specimens is extensive and there is only space in this short

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review to consider a few examples to illustrate the main features of this field. Photoelectron spectroscopy has made major contributions to the understanding of the electronic structure and magnetism of impurities in elemental metals. In particular the correct description of the electronic structure of impurities of transition metals in noble and alkali metals are long standing problems that impact on the magnetic behaviour, electron specific heat and electrical conductivity of these materials. References to the extensive early reviews of this subject and a detailed study of the electronic structure of Mn impurities in noble metals can be found in the papers by van der Marel et al. [52,53]. In these studies electron spectroscopy techniques were used to determine the electronic structure of both the minority and majority spin d states of Mn in Cu and Ag and the results were compared with the results of model calculations. The results question the validity of the commonly used Kondo Hamiltonian and show that hybridisation between the Mn states and those of the host can introduce new exchange mechanisms not foreseen in conventional models. The capability of the commonly used Friedel–Anderson [54] virtual bound state (VBS) model to describe the electronic structure of Ni impurities in Cu, Au and Zn and for Co impurities in Au metals has been investigated using UPS [55] and of Pd impurities in the noble metals and of Pt in Ag by XPS [56]. It is found that crystal field and spin–orbit splittings make additional contributions to the width of the VBS in some cases. It was also found that the VBS model fails to predict the position and widths of photoemission measurements of the d states of impurities in simple metals [57]. Such spectra are often more accurately described as bonding states between the d band impurity and the host and can be interpreted from the results of band structure calculations [57–59]. van der Marel et al. [60,61] also studied the changes in the density of states of Cu, Ag and Au induced by Pd and Pt impurities and compared the results of XPS and UPS measurements with calculations of the impurity densities of states using a modified version of the Clogston–Wolff model Hamiltonian. They also considered the influence on the electronic structure of local lattice relaxations and showed that their neglect can lead to qualitatively incorrect predictions of the local density of states. Some of the earliest applications of both XPS [62] and UPS [63,64] were in studies of the conduction bands of concentrated alloys. Fuggle et al. [65] measured the XPS of the conduction bands of 60 Ni and Pd alloys with 20 different elements. The results were used to rationalize the results of specific heat and magnetic measurement on a number of these alloys. The studies referred to above and many others employed XPS or UPS techniques with fixed energy laboratory sources. While it is possible to make significant advances with such techniques they suffer from the disadvantage that it is usually not possible to separate the contributions of the individual elements to the total density of states of the alloy. A major advance was the realisation that it was possible to make such a separation by using the tuneable nature of synchrotron radiation to exploit the variation in the elemental photoelectron cross-sections with photon energy, and in particular the existence of a Cooper minimum in the photoelectron cross-sections for the d levels of some elements. Theoretical results show that the photoelectron cross-sections of atomic orbitals that have a radial node go through a minimum at some photon energy [8]. Experimental measurements [9] show that the detailed dependence of the photoelectron cross-section on photon energy and the precise energy of the minimum can be strongly influenced by solid state effects [10,66] as shown by Cole et al. [14,67] who found a remarkable difference in the photon energy dependence of the photoelectron cross-section of Ag impurities on the host metals Al and Cd (Fig. 1). The Cooper minimum depends crucially on cancellations in matrix element integrals and this makes the photon energy dependence of the photoelectron cross-section for such systems

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this has given rise to considerable controversy [3,12,72–75] which, while focussed on this particular alloy system, has implications for the analysis of results obtained on many similar alloys. This controversy is discussed in detail later. Finally we note that the information on the density of states of disordered alloys deduced from electron spectroscopy can provide insight into other aspects of the electron structure of these systems as demonstrated by Nahm et al.’s [72] resolution of discrepancies in the interpretation of the electron specific heat of NiPt alloys through a comparison of experimental and theoretical results for the Ni density of states at the Fermi energy in these systems.

4. Electron correlation: the Hubbard U

Fig. 1. The photon energy dependence of the photoelectron cross-section of the Ag 4d states from experiments on Cd97 Ag3 , Al95 Ag5 [14], Ag metal [68] and Ag on Si(1 1 1) [69]. Reproduced from [67] with permission.

very sensitive to changes in the elemental wave functions. Fig. 1 shows that for Al95 Ag5 the photon energy dependence of the photoelectron cross-section of the Ag 4d states is very similar to that of Ag atoms in gas phase [70] and isolated Ag atoms on the Si(1 1 1) surface [69]. This suggests that the Ag d states of the essentially isolated Ag impurities are not significantly influenced by hybridisation with the sp band of the Al host. In Cd97 Ag3 however the Ag 4d states are clearly significantly distorted by overlap with the Cd 4d states even though the Cd d band is quite bound (Fig. 2) and only a small percentage of the intensity of the Ag 4d virtual bound state is mixed into the Cd d band. The effect of this distortion on the photoelectron cross-section is even greater than that observed in Ag metal (Fig. 1). By exploiting Cooper minima it has been possible to get a good idea of the separate elemental contributions to the total densities of states of a variety of disordered binary alloys: AuNi [11], AuFe [22,13], Ag impurities in Al and Cd [14,67], CuAu [15,71], Au95 Pd5 [16], NiPt and CuPt [17,22], AuAg [18], Pt3 V and Pt3 Mn [19], AuPd [20] and even the three component Mn based Heusler alloys [21]. One of the most interesting cases is that of the CuPd system and

Fig. 2. Photoelectron spectrum of Cd97 Ag3 [14]. The Ag VBS is at ∼6 eV. The Cd d band is ∼10–13 eV. Reproduced from [67] with permission.

Theories of the electronic structure of solids usually adopt an independent particle approach the major deficiency of which is the neglect of the interactions between the electrons which are expected to correlate their motions. This deficiency is usually overcome to some extent by including the effects of electron correlation in an average way, most notably in density functional theory (DFT) [77–79] in which an exchange correlation potential evaluated from the charge density is included in the Hamiltonian. However the assumed independent motion of electrons implies significant charge fluctuation (i.e. “snapshots” would reveal departures of the local electron count from the average) even though the corresponding Coulomb interactions are not accounted for. Many years ago Hubbard [80–82] showed that important aspects of the electronic structure of solids such as magnetism and electrical conductivity can be strongly influenced by such local Coulomb interactions. In the Hubbard Hamiltonian a Coulomb energy, U, for electrons at the same site is added to the usual energy band term. When U is large relative to the independent electron band-width, W, it has the effect of localising electrons, giving rise to the highly correlated Mott–Hubbard insulating state. The Hubbard model can only be solved exactly for model systems, however considerable progress has been made by semi-empirical approaches in which an empirical U is “added” to the electronic structure derived from an independent particle approach. In recent years a quite promising route to the solution of this problem appears to be a combination of DFT with dynamical mean field theory (DMFT) [83]. It is anticipated that the application of DFT + DMFT to disordered and correlated systems will be fruitful but rather demanding in its implementation. Major advances in the determination of on-site electron correlations energies have been made using electron spectroscopy techniques and experimental studies of disordered alloys have been at the forefront of these developments. The key advances have been made in experimental and theoretical studies of Auger spectra since the localised atomic wave function of the initial core-hole ensures that the two holes created in the final state are initially localised on a single atomic site in a solid. When the two final state holes involve valence states then their subsequent fate will be determined by the ratio U/W. The Auger profile thus contains information on the value of the Hubbard U. The theory of Auger profiles was developed independently by Cini [84,85] and Sawatzky [86] and provided an explanation of Powell’s observation [87] that the while the core–valence–valence (CVV) Auger profile of Al metal resembled the self-convolution of the single valence band density of states, as suggested by Lander [88], the CVV profile of metallic Ag, where the valence holes are in the Ag d band, resembled an atomic profile. Cini and Sawatzky developed the theory in slightly different ways and it is useful to compare their approaches to the much simpler system of a diatomic molecule [89] in which there are two important Coulomb interactions between the final state holes created in an Auger process; the on-site term Uaa where both holes are located on one atom and Uab where the holes are localised on different atoms. Sawatzky

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[86] developed his analysis from the Hubbard Hamiltonian while Cini made use of the Anderson Hamiltonian. In Sawatzky’s approach there is a Uaa on all the atomic sites and the dispersion of the double-hole states is included. Cini’s approach can be seen as a generalisation of the treatment of the diatomic molecule in which all the on-site terms are zero except for the active site Uaa . The relationship between the two approaches has been clarified [90,91] and the subject has been reviewed by Verdozzi et al. [92]. It is important to note that both approaches assume that the interaction between delocalised holes, Uab in the diatomic molecule, is zero and lead to the same result namely that when Uaa > W a split-off state is formed that contributes a narrow, atomic like, state to the Auger spectrum. The first experimental proof that Cini–Sawatzky (CS) theory could explain the shape of Auger profiles was provided by studies of the M4,5 N4,5 N4,5 Auger profile of Ag in MgAg and AlAg alloys [93,94]. By varying the alloy composition it was possible to vary the single electron band-width W which could be determined by photoemission. By combining measured values of W with atomic structure calculations of the Ag M4,5 N4,5 N4,5 Auger profile [95] it was possible to explain the dependence of the shape of the Auger profile on the alloy composition in terms of CS theory. The key finding was that the dependence of U on the LSJ term structure of the final state holes gives rise to a variation in the ratio U(LSJ)/W across the Auger spectrum. As a result only some of the final state core-holes satisfy U(LSJ) > W and are localised on-site and give rise to an atomic profile. For smaller values of U(LSJ) these LSJ components are delocalised and give rise to a broad contribution derived from the self-convolution of the single electron density of states but distorted by a factor determined by the ratio U(LSJ)/W. It is possible to calculate values of the on-site U together with d band densities of states from first principles for elemental metals [96] though the calculations have not yet been extended to alloys. Auger profiles that create final states in narrow d bands in alloys can now be interpreted in terms of CS theory and can be used to determine values of the Hubbard U(LSJ) [24,97]. An analysis [26,98,99] of the very different Pd M4,5 N4,5 N4,5 Auger profiles observed from a range Pd alloys; Al80 Pd20 , Mg75 Pd25 , Cu95 Pd5 , Ag80 Pd20 and Ag95 Pd5 , yielded values for the 1 S0 ,1 G4 and 3 F4 LSJ terms of the final state holes, U(1 S0 ), U(1 G4 ), U(3 F4 ), of 5.4, 3.0 and 1.1 eV, respectively. Whereas for individual alloys the values could only be determined to an accuracy of between ±0.2 and ±0.5 eV the spread in values obtained across all the alloys was ±0.2 eV confirming Hubbard’s original idea that, provided one allows for dependence on the term structure, U is essentially an on-site quantity. We will return to this point below. In alloys between simple metals and d band metals the d band density of states can usually be determined directly by photoemission. However for alloys between d band metals the d bands mix and it is more difficult to determine the density of states of one elemental component even by exploiting the Cooper minimum. The Auger profiles that create two-hole states in an elemental d band are very sensitive to the d band density of states and can act as a check on the accuracy of both theoretical and experimental results for the density of states and provide accurate results for the on-site correlation energy, U(LSJ). This is illustrated in Fig. 3 which a shows a comparison between experimental results for the Ag M4,5 N4,5 N4,5 Auger profile of Cd97 Ag3 and theoretical profiles deduced from the CS formalism [100]. The Ag d density of states in this alloy forms a VBS (Fig. 2) in the sp band of the host and with a small admixture into the more bound d band of Cd. The calculated Auger profile is sensitive to this admixture between the Ag and Cd d states and is consistent with the 3.2% admixture predicted by the Clogston–Wolff model [100]. The Auger profile is also very sensitive to the on-site electron correlation energies U(LSJ) as shown in Fig. 3. The LSJ term splittings can be considered fixed [95] and the profiles shown in Fig. 3

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Fig. 3. The crosses are the experimental spectra of the Ag M4,5 N4,5 N4,5 Auger spectrum of Cd97 Ag3 . The full curves are theoretical spectra calculated using the Cini–Sawatzky formalism with a range of values of U(1 G4 ) in eV. The splittings of the other LSJ components are fixed with respect to U(1 G4 ) [95]. Reprinted from [100] with the permission of IOP Publishing.

are calculated for a range of values of U(1 G4 ). Good agreement with experiment is obtained for U(1 G4 ) = 4.1 ± 0.2 eV. CS theory predicts that a single value of U(LSJ) determines both the kinetic energy and the line shape of the contribution of the LSJ component of the two-hole final state to the Auger spectrum. It is important to note that the work described above, which confirmed the essential features of the CS theory, concentrated on deriving the shape of the Auger profiles from their predicted dependence on the ratio U(LSJ)/W and did not give a detailed comparison of line shape with kinetic energy. This was an unfortunate omission since it later became clear in a study of the N6,7 O4,5 O4,5 Auger spectrum of metallic Au [101–103] that one set of values of U(LSJ) did not give simultaneous agreement with the Auger profile and the kinetic energies of these transitions. There is a small discrepancy in that whereas LSJ term separations related to a value for the most intense component U(1 G4 ) of 3.4 eV reproduced the Auger profile they predicted the kinetic energy to be 1.2 eV higher than observed. A value of U(1 G4 ) of 4.6 eV was required in order to reproduce the observed kinetic energy but this did not reproduce the Auger profile. There is no set of values of the U(LSJ) that is able to provide simultaneous agreement with the kinetic energy and the profile of these tran-

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sitions. A similar discrepancy was later found for the M4,5 N4,5 N4,5 Auger transitions of metallic Ag [104]. The source of this discrepancy can be traced to the neglect in both the Cini and Sawatzky versions of the theory of the off-site interaction which is represented by the Uab term in the treatment of the diatomic molecule [89]. While a U restricted to a single site is a convenient assumption it has been suggested that even this on-site interaction is influenced by more distant electron–electron interactions [105] and it is reasonable to suppose that the interaction will show a slight dependence on the separation, R, between the two holes as suggested by Verdozzi et al. [103]. The inclusion of such a dependence does resolve the discrepancy between the predicted dependence of the Auger profile and Auger kinetic energy on U(LSJ) for both metallic Au [103] and metallic Ag [104]. Verdozzi and Cini [106,107] represent the off-site interaction by

UR = UıR0 + AR exp

 −R  

mation has been obtained from two quite different approaches: the analysis of Auger parameters and the analysis of Auger profiles. While both approaches exploit Auger spectra they do so in very different ways. The analysis of Auger parameters concentrates on comparisons between core-level binding energies and the kinetic energy of Auger transitions in which all three states involved in the transitions are in the atomic core. The kinetic energies of such core–core–core (CCC) Auger transitions show the same sensitivity to the potential in the atomic core as the kinetic energies of photoelectrons ejected from core-levels. The analysis of Auger profiles is concerned with the relationship between the line shape of Auger transitions that involve final state holes in the valence levels and the shape of the local valence electron density of states. These Auger processes are of two types core–core–valence (CCV) transitions in which only one of the final state holes is in a valence level and core–valence–valence (CVV) transitions in which both final state holes are in valence levels.

(1)

where the on-site interaction produced by the delta function is augmented by a Thomas–Fermi potential with amplitude, A, and screening length, , which is restricted by  R to the range 0 < R < 2a where a is the lattice constant. Beyond 2a the holes are considered to be delocalised and U = 0. However this extension to the on-site term has not yet been included in the analysis of the Auger profiles of disordered alloys. There is also a potential problem in that a full analysis of a disordered system will require a determination of the local density of states on-site rather than the average density of states that has been employed so far. Abrikosov and Johansson [108] showed in a theoretical treatment that the local densities of states in CuPd, CuAu and CuZn alloys changed significantly with variations in the composition of the shell of nearest neighbours round a site. In Ref. [109] an investigation was performed of the one and two-body local densities of states in systems with correlation and disorder. The method was based on a minimal-super-cell approach, through a selection of a quasi-random supercells, in which constraints were imposed on the statistical/experimental occurrence ratio of quantities such as concentration, local shell coordination, etc. Such an approach is amenable to an ab initio simultaneous treatment of disorder and correlation. A qualitative discussion was carried out for random binary alloys of the late transition metals in terms of a disordered Hubbard model. Results for the local densities of states, and their dependence on the local environment, showed how disorder and electron correlation together induce a transfer of spectral intensity in energy regions different from those in ordered systems. It was also suggested that CVV Auger spectra could be spatially resolved, thus becoming sensitive to local disorder, via coincidence Auger spectroscopy. This last procedure is beginning to be within the reach of present day experiments [110–114] as will be discussed in a later article [42]. The CS theory described above and its extension to include offsite interactions is limited to systems in which the d band is filled [115–118]. There have been a number of attempts to extend the theory to incompletely filled d band and these have been reviewed by Verdozzi et al. [92]. Provided the d band is almost completely filled good agreement can be obtained between theory and experiment as shown by Cini and Verdozzi for Pd metal [118].

5. Electron screening, charge transfer and s–d hybridisation Electron spectroscopy techniques have also provided information on electron screening, charge transfer and the hybridisation between s and d valence electrons in disordered alloys.This infor-

5.1. Auger parameters The Auger parameter was introduced by Wagner [30,31] to avoid the dependence of measurements of the core-level binding energies between different materials on the calibration of electron spectrometers. By defining the Auger parameter, ˛, as the difference between the kinetic energy of a feature in the Auger spectrum and a core-level photoelectron line ˛ = KE(CCC) − KE(C)

(2)

Wagner derived a parameter that was independent of the calibration of the spectrometer. Provided the energy scales of electron spectrometers are linear then measurements of elemental Auger parameters in different materials on different instruments can yield reference free insights into the dependence of the local electronic structure on material composition. One of the early successes of the analysis of Auger parameters was to provide an explanation for observations that the core-levels of the two elements in some binary alloys showed the same shift in binding energy with respect to the core-hole binding energies in the pure elements [119]. Such observations demonstrated that the binding energies of core-holes did not just depend on the local atomic potential since any charge transfer that occurred on the formation of the binary system could not be in the same direction with respect the potential in the atomic cores of the elemental materials for both elements. Clearly there was also a final state contribution to core-level binding energies that arose from electron screening of the final state core-hole. The analysis of Auger parameters, which have been defined in a number of ways, has made a major contribution to the separation of initial and final state effects in electron spectroscopy on a wide variety of materials and this has been reviewed by Moretti [32]. In metals one expects almost perfect screening in the final state due to the mobility of the conduction electrons and this makes it possible to introduce some simplifying assumptions into the analysis of Auger parameters. In particular this has made it possible to investigate the role of charge transfer in the initial state in alloys. We should note immediately that charge transfer is a controversial topic since the concept is not a quantum mechanical observable and indeed even if one knew the complete electron distribution in a material it would not be possible to associate particular charges with particular atoms due to the indistinguishability of electrons. However core-level binding energies and CCC Auger kinetic energies are determined in part by the potential in the core of an atom and this is a quantum mechanical observable and one that can be investigated by Auger parameter measurements. By concentrating on changes in atomic core potentials and by isolating initial and final state effects through measurements of Auger parameters it has been possible to make considerable progress in quantifying charge

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transfer in alloys. The analysis of Auger parameters has been particularly useful in clarifying the results of studies that combined core-level binding energies measurement with Mössbauer isomer shifts [120–128]. A key advance came through studies of Auger parameters in AuZn and AuMg alloys [33]. These systems together with AuCs had been studied by Wertheim et al. [126] who had measured both corelevel binding energies and Au Mössbauer isomer shifts, the latter providing insight into the s density of states at the nuclei. They concluded that there was significant charge transfer in the initial state of these materials to the Au site of the order of 0.25 e for AuZn, 0.4 e for AuMg and 0.7 e for AuCs and that, contrary to previous suggestions [120,125], there was no back transfer of d electrons from the Au to the other element of the alloy which might have partially neutralised the large transfer of electrons into the 6s orbitals of Au. The situation was clarified [33] through the inclusion of Auger parameter measurements in the analysis and the recognition that the Auger parameter depends on the changes in the core potential when an inner shell electron is removed. For metals the initial and final state contributions to this change in potential can both be represented by on-site terms since the screening is local. This makes it possible to represent the difference in Auger parameter, ˛, between two environments by a potential model leading to the expression

   dk   i

˛ = 

i

qi

dN

+ ki − 2

dki dN

  dq  i

dN

(3)

where the qi are the local atomic charges arising from the electrons in the valence orbitals i, ki is the change in core potential when a valence electron is removed from orbital i and N is the number of core electrons. In metals dqi /dN is one since exactly one screening electron is attracted to a core-ionised site from the conduction band. The parameters ki and dki /dN can be determined from atomic Dirac-Fock calculations. An important finding from the Dirac-Fock calculations was that these parameters are significantly different for s and d valence electrons [33]. The analysis of Auger parameter data for AuZn and AuMg using (3) made it possible to derive local valence populations that satisfied both the Mössbauer data for these systems and AuCs and the electron spectroscopy measurements. It was found that the transfer of charge into the 6s electrons of Au in the alloys was partly balanced by a back transfer of 6p electrons a conclusion that was supported by the results of electronic structure calculations for AuCs [129]. The net charge transfer was found to be quite small, 0.1 e and 0.2 e to the Au in AuZn and AuMg, respectively. The potential model described by (3) requires elemental parameters derived from atomic structure calculations [130]. It has been applied to many systems [131,132] and gave rise to a new scale of electronegativity [27,133] putting this important chemical concept on a firm quantum mechanical footing. In metals it has been particularly useful in separating the contributions of sp and d screening in the final state [134,135]. In disordered alloys there are many factors that can influence the variation of Auger parameters with alloy composition as illustrated by the extensive studies of the CuPd alloy system by several groups [34–38,40,41,75,136–141]. To begin with since core-level binding energies are usually measured relative to the Fermi level it may be necessary to make corrections for the variation with composition of the position of the Fermi energy in the conduction band [136,142]. Furthermore low energy many body excitations endow core-level photoelectron lines with an asymmetry determined by the local density of states at the Fermi energy [143,144]. As a result the photoelectron peak maxima are displaced from their “true” positions [34]. In the CuPd system for example the asymmetry of the Cu 2p core-levels is small and independent of alloy composition suggesting that the Cu 3d band density of states lies below the

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Fermi level for all compositions, in agreement with the results of electronic structure calculations [136] and with valence band photoelectron spectroscopy [75]. However the asymmetry of the Pd 3d photoelectron line is strongly dependent on alloy composition since the Fermi level lies in the Pd d band for Pd concentrations above 40% and it is necessary to make corrections for the dependence of the core line asymmetry on composition in determining core-level binding energies for Pd [34]. In measurements of Auger parameters it is important to choose CCC transitions that are deep in the core of the atom. In a CuPd study [38] the Cu L2,3 M2,3 M2,3 [145] and Pd L3 M4,5 M4,5 [146] were used and the analysis of the Auger parameters showed that for both elements in the CuPd system final state effects are small. In interpreting the dependence of the Auger parameters on alloy composition it is important to adopt generalised expressions (3) which allow for the different contribution made to the core potentials by valence electrons of sp and d character [137,147], to consider the relationship between the zero of potential in models [36,148,149] and in the experimental energy scale and to take account of a number of factors that influence the values of parameters to be used in potential models [120,125,130,150]. After taking all these factors into account it was possible [38] to interpret the results in terms of an initial state charge correlated model (CCM) [40,138–140] that had been developed for ab initio studies of the energy of disordered alloys and which will be described in a later article [42]. It was found that core-level shifts were consistent with the average electrostatic potential of the random CCM lattice given by the degree of charge transfer predicted for the CuPd system by the ab initio calculations [40,138–140]. An alternative potential model for the interpretation of Auger parameter measurements to that described above was developed by Kleiman and co-workers [151–155]. This approach starts from the Kohn–Sham equations [156] but derives an expression rather similar to (3). It has been applied to a wide variety of systems including CuPd [152], CuPt [153], AuPd [154], and the AuZn and AuMg systems [147]. In general this approach yields charge transfers in disordered alloys that are smaller those found using models derived from (3); of the order of hundredths of electrons per atom. This approach has been described and reviewed in detail by Kleiman and Landers [151] who also compare the assumptions that are made with those made in the alternative approach derived from Ref. [33]. The difficulties in applying such models to the analysis of Auger parameter measurements is captured in the title of the Kleiman and Landers paper [151] “Energy shifts and electronic structure changes in alloys: an unfulfilled promise?” It is clear from the discussion above that while the measurement of Auger parameters can provide considerable insight into the electronic structure of disordered alloys it is difficult, though possible by making use of empirical models, to establish a clear relationship between the experimental measurements and the results of first principles calculations. Fortunately there has recently been significant progress in establishing a link between the results of experimental measurements and ab initio calculations of the electronic structure of disordered alloys through the development of the complete screening picture [157–159] which provides a consistent scheme for evaluating both initial and final state contributions to core-level binding energies and Auger kinetic energies measured by electron spectroscopy. In a study of AgPd alloys it was shown [39] that first principles calculations in the complete screening picture gave good agreement with corelevel binding energies of the Ag 3d5/2 levels of Ag and Pd and the L3 M4,5 M4,5 of both elements across the whole composition range. The analysis of the theoretical results provides considerable insight into the electronic structure of the ground state and the mechanisms of electron screening in excited states in AgPd alloys. This approach establishes a sound methodology for linking

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the results of electron spectroscopy measurements and first principles theoretical treatments of the electronic structure of disordered alloys. 5.2. Auger profiles and electron screening We have described how the analysis of the Auger profiles of CVV transitions can yield insight into the local d density of states and the local electron correlation energy. When the final hole states created in such transitions do not involve d bands the profiles of CVV transitions are not usually influenced by the correlation effects captured by the Cini–Sawatzky theory [84–86] and its extension to off-site interactions [103]. This is because for such transitions U is usually zero and certainly very much less than W and the Auger profiles resemble a self-convolution of the local density of states as originally suggested by Lander [88]. This is also true for CCV transitions with the important caveat that the local densities of states in both the initial and final state of these transitions involve a core-hole. This gives rise to an important difference between the profiles of CCV and CVV transitions in that in the former the two-hole final state reflects the local density of states in the presence of a corehole while the latter it corresponds to the local density of states in the presence of two valence holes. An important simplification is that due to the extended nature of valence states of s and p character they are expected to delocalise rapidly and to only weakly perturb the on-site potential and consequently it is reasonable to suppose that the densities of states probed in CVV transitions is very similar to that in the ground state of the material. These considerations are captured in the final state rule [160–164], which applies not only to Auger transitions involving valence levels but also the profiles of X-ray emission involving valence levels, and concludes that, provided U can be neglected, the shape of the s and p contributions to the Auger profiles of CCV and CVV transitions are given by the shape of these densities of states in the final state while the intensities of these contributions are given by the local valence electron configuration in the initial states. The theoretical underpinning of the final state rule has been described several times [160–164] and expressions have been given relating the profiles of CVV transitions to the local valence electron configurations in the initial and final states, the appropriate matrix elements and the local densities of s and p character in the ground state Ds (E) and Dp (E), respectively [27,165]. Unfortunately CVV transitions are very weak and as far as we are aware it has only been possible to make accurate measurements of the profile of these transitions for a few first row elements [166,167]. However the insight obtained into the local valence density of states in these systems demonstrates that they have the potential to become a powerful probe of local densities of states (LDOS) in disordered alloys since they are element specific and, apart from “accidental” overlaps, they will occur in regions of the electron spectrum free from the contributions of other elements. There has been a considerable amount of research on the more intense CCV Auger profiles of disordered alloys. The final state rule yields the following expression [27,165] for the profiles of these transitions 2 2 ACCV (E) = MCCs Ds1 (E) + MCCp Dp1 (E)

The final state rule was established by Von Barth and Grossman’s analysis of the KLV transitions of Na metal [161,162]. They showed that the screening of a core-ionised site distorts the shape of the ground state density of states of local s character, Ds (E), in a simple metal, by inducing a peak at the bottom of the band. This final state density of states is denoted Ds1 (E). However the p contribution to the screening charge is more evenly distributed across the band and the shape, though not the intensity, of Dp1 (E) is similar to the local p LDOS in the ground state, Dp (E). These considerations explained why the KL1 V Auger profiles of the simple metals Na, Mg and Al are composed of two peaks with varying intensities determined by the value of the matrix elements, one arising from Ds1 (E) and one from Dp1 (E). The roughly equal intensity of the two contributions to the KL1 V profiles arises from the fact that the matrix elements for KL1 V transitions in (4) are roughly equal. This contrasts with the situation for the KL2,3 V transitions in which the intensity ratio MCCs :MCCp is roughly 1:5 with the result that the KL2,3 V profiles consist of a strong contribution from Dp1 (E) with a weak contribution from Ds1 (E) to low kinetic energy. This difference in the sensitivity of the KL1 V and KL2,3 V profiles to Ds1 (E) and Dp1 (E) means that a superposition of the two profiles yields a crude but direct way of studying how the LDOS of s character varies with the atomic environment in alloys. Fig. 4 shows such a comparison for the KLV spectra of Mg in the elemental metal and in alloys with Li and Al [168]. For the pure element the difference in the two profiles, showed by the shaded region, reveals the distortion in Ds1 (E) required to screen the core-hole in the final state as expected from theory [161,162]. The figure, and a more detailed comparison of the profiles of both Mg and Al KLV Auger and L2,3 V and KV X-ray emission profiles in a wide range of AlMg alloys [169], shows that the shape of Ds1 (E) is very dependent on the alloy composition. This dependence can be understood by combining the final state rule with the equivalent cores approximation. If we assume that Mg metal consists of an array of local Mg2+ ions in a sea of conduction electrons then a KLV transition will create a local Mg3+ ion that will be screened by attracting an additional electron from the conduction band

(4)

For the CCV transitions the local configuration is the same in the initial and final state and in this expression the MCCs and MCCp are matrix elements and the amount of local valence s and p charge, which by the final state rule will be the same in the initial and final states of these transitions, is implicitly included in the unnormalised LDOS around a core-ionised site Ds1 (E) and Dp1 (E). We note that these LDOS’s cannot be obtained from ground state band structure calculations but require a calculation for a core-ionised site.

Fig. 4. The full lines show the KL1 V spectra and the dashed lines the KL2,3 V spectra of Mg in each material. Due to matrix element effects in Eq. (4) the profile of the KL1 V transitions is determined by roughly equal contributions from the local s and local p densities of states while the intensity ratio of the local s and p contributions to the KL2,3 V profile are in the ratio 1:5. As a result the superposition of the two profiles reveals the shape of the local s densities of states. The diagonal shading is the difference between these profiles showing the distortion of the s local density of states in the different atomic environments. The energy scale is referenced to the Fermi energy in eV. Reprinted from [174] with permission.

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into the local Ds1 (E) and Dp1 (E) densities of states. Ds1 (E) will be distorted from the ground state Ds (E) but Dp1 (E) will have roughly the same shape but possibly an increased intensity compared to Dp (E). For Mg dilute in Al the Mg2+ ion will be in an environment of Al3+ sites. A Mg KLV transition will thus create an Mg3+ ion in an environment of Al3+ sites and it will not be necessary to provide additional screening of this site with the result that the difference in profiles of the KL1 V and KL2,3 V profiles will be small (Fig. 4). For Mg dilute in Li the Mg2+ ion will be in an environment of Li+ sites and the creation of an Mg3+ ion by the Auger process will require the attraction of two conduction electrons to screen the site giving rise to significant distortion of Ds1 (E) and possibly the formation of a bound state. Such studies have provided a stimulus for theoretical work on the local electronic structure in disordered alloys [170,171]. An important point is that although the measurements are on excited states they provide insight into the local electronic structure in the ground state since by the equivalent cores approximation the local density of states around a core-ionised element of atomic number Z will resemble the local density of states in the ground state of element Z + 1. This expectation has been confirmed in the detailed the study of MgAl alloys referred to earlier [169]. The profiles of CCV transitions also provide insight into local screening in alloys between simple metals and d band metals and into the effects of the hybridisation of d bands with s and p bands [172–175]. In an alloy between a simple metal and a d band metal one expects that the d band will hybridise with the s and p bands of the simple metals and give rise to a dip in the local densities of states of s and p character roughly centred on the energy of the d band. This will change the profile of the local s and p densities of states screening the core-hole in CCV transitions. This is confirmed by comparisons of the valence band photoemission and Mg KLV Auger profiles of Mg alloys with Ni, Cu, Zn, Pd and Ag with the Al KV Xray profiles of corresponding Al alloys with these d band elements [172,173]. A key finding is illustrated in Fig. 5 [174], the lower panel of which shows that the Zn d band in Mg7 Zn3 is below the bottom of the Mg sp band and consequently the d band does not influence the local Dp1 (E) screening the final state core-hole on a Mg site as monitored by the profile of the Mg KL2,3 V transitions. As the d band moves closer to the Fermi level in MgCu2 a dip appears in Dp1 (E) at the position of the Cu d band. For MgNi2 the d band is very close to the Fermi level and the hybridisation effect can just been seen at the low binding energy side of the Mg Dp1 (E) distribution. Similar results though with interesting differences in detail are found for other alloy compositions and for MgPd and MgAg alloys [172,173]. While it is expected that the Mg KL2,3 V profile will not be influenced by the tightly bound d band in Mg7 Zn3 it does not follow that this will also be true for the Mg KL1 V profile since as we have seen the presence of a core-hole on a Mg site induces a strong distortion of the local Ds1 (E) which gives rise to a peak at the bottom of the band. However the low energy peak in the Mg Ds1 (E) is reduced in intensity in Mg7 Zn3 and disappears altogether in MgZn [169]. This is an interesting result because it implies that the screening of a core-hole on a Mg site in MgZn is accomplished by polarising the d density of states which is expected to be localised on Zn sites. Similar conclusions on the role of screening by d bands follow from comparisons, as a function of alloy composition, of the relative intensities of features in the local Ds1 (E) and Dp1 (E) “above and below” the d band in other alloys [165].

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Fig. 5. The bottom panel shows the XPS spectra of the conduction bands. The top panel shows the corresponding Mg KL2,3 V Auger profiles. The energy scale is references to the Fermi energy in eV. Reprinted from [174] with permission.

6. CuPd and AgPd alloys The CuPd and AgPd alloy systems form substitutionally disordered face centred cubic (FCC) solid solutions across the whole composition range. The CuPd system shows ordered phases at some compositions though the transformations to the ordered phases

require careful thermal treatments of the specimen. There are no ordered phases in the AgPd system. These systems have long been the focus of CPA calculations, an approach that originally treated the individual elements as an average scattering potential in a sin-

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gle site approximation in a perfect FCC lattice. A good place to start is with the self-consistent-field Korringa–Kohn–Rostoker coherent potential approximation (SCF-KKR-CPA) calculation of Winter et al. [136] for CuPd since this work provides a good introduction to CPA techniques and provides references to previous work. The calculations predicted that for the complete range of alloy compositions the Cu and Pd densities of states are thoroughly mixed in a common d band. A very different situation prevails in AgPd alloys where the results of SCF-KKR-CPA calculations [176] predict that the Ag and Pd densities of states in these alloys are split into two different bands. For AgPd alloys the theoretical results are a good agreement with experimental results for the photoemission obtained from the d bands over the whole range of alloy compositions [99,177]. Furthermore the Pd density of states given by the calculations also gives reasonable agreement with experiment when included in CS type calculations of the Pd M4,5 N4,5 N4,5 Auger profiles [178]. The agreement with the experimental Pd M4,5 N4,5 N4,5 Auger profiles in Ag rich AgPd alloys could be improved by modifications to the Pd density of states [179,180] and this lead to some disagreement over the density of states of Pd impurities in Ag [181] that was eventually resolved [182]. In addressing the controversy that has arisen over the interpretation of experimental and theoretical results obtained for the CuPd alloy system [3,12,72–74] it should be noted that surface segregation is not a problem in CuPd alloys [12,48] and it is one of the easiest systems in which to separate the elemental contributions to the joint densities of states since while Pd has a Cooper minimum Cu does not. Consequently by measuring the photoelectron spectrum at the Pd Cooper minimum one obtains the Cu density of states and the subtraction of this from the joint density of states measured when the Pd photoelectron cross-section is a maximum yields, after allowing of photoelectron cross-sections, the Pd density of states. For Cu75 Pd25 this approach revealed an experimental density of states for Cu that was in very good agreement with the results of the SCF-KKR-CPA calculations [12]. The calculations predicted that for the complete range of alloy compositions the Cu and Pd densities of states are thoroughly mixed in a common d band and that contrary to previous interpretations of XPS results [34,183,184] a Pd virtual bound state is not formed in Cu rich alloys. Measurements of the Cu L2,3 -M4,5 XES from a Cu10 Pd90 alloy, used as the anode in a photoelectron experiment [185], confirmed that the Cu density of states did indeed stretch over the whole d band at low Cu concentrations as predicted and later XPS experiments were consistent with the predicted common d band in CuPd alloys [186]. The calculations also predicted that the Pd d band was filled for Pd concentrations lower than 40%, a result supported by BIS measurements of the empty density of states [187]. However it was hard to reconcile the SCF-KKR-CPA results for the Pd DOS with the results of UV excited photoelectron spectroscopy [60] and with the results of the photoemission exploiting the Pd Cooper minimum [12]. These results showed a significant disagreement with the theoretical density of states by indicating that the calculations significantly over estimated the Pd density of states at the bottom of the d band (Fig. 6), a conclusion confirmed by XES [188] and supported by disagreements between experimental and calculated Auger profiles of the Pd M4,5 N4,5 N4,5 Auger transitions [178]. It was suggested [12] that the discrepancy between theory and experiment for the Pd density of states might arise from the neglect of local lattice relaxation around the Pd sites as indicated by the work of van der Marel et al. [60] and shown to be significant for the CuPd system by EXAFS measurements [3,4]. However Nahm et al. [72] argued that the disagreement between the experimental and theoretical results shown in Fig. 6b arose from the neglect of the energy dependence of the photoelectron matrix elements and an energy dependent self-energy both of which act to reduce the measured intensity of the Pd density of states at the bottom of

Fig. 6. The dots show the empirical densities of states for Cu (a) and Pd (b) in Cu75 Pd25 determined by making use of photoemission experiments exploiting the Cooper minimum in the Pd 4d photoelectron cross-section [12]. The full curves are the theoretical results given by SCF-KKR-CPA calculations [136]. Reprinted (Fig. 5) with permission from [12]. Copyright (1987) by the American Physical Society.

the d band. The difference between the two interpretations can be found in [73,74] and Nahm et al. developed their view further in a later work [75]. They concluded that lattice relaxation effects do not have an important effect on the Cu and Pd densities of states in concentrated CuPd alloys. While it is clear that photoelectron matrix elements and self-energy effects do contribute to the determination of elemental contributions to the density of states in this and other alloy systems the methodology adopted by Nahm et al. [72,75] is not sufficiently robust to support their conclusion that local lattice relaxations do not have an important influence on the local densities of states in alloys in which there is a significant mismatch in the size of the constituent elements. In particular in estimating the contribution from matrix element effects they normalise their results using experimental results for the intensity of elemental d bands. Unfortunately this approach does not allow for the remarkable sensitivity of Cooper minima to the mixing of d bands [14,67] (Fig. 1) which would seem to make it impossible to make a realistic empirical determination of matrix elements for Pd in CuPd. We suggest that an accurate result for the Pd density of states in this, and probably related alloys, cannot be easily obtained from photoemission experiments.

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We note that a number of theoretical approaches indicate that local relaxations do have an important influence on local densities of states in alloys and in the CuPd system they act to reduce the discrepancy between theoretical and experimental results for the intensity of the Pd d states at the bottom of the d band shown in Fig. 6b [189–191]. A more stringent test of the energy distribution of the Pd density of states in CuPd is the profile of the Pd M4,5 N4,5 N4,5 Auger transitions. The localisation of the initial 3d core-hole state on a Pd site means that these transitions create two-hole final states, 4d2 on a Pd site. The Auger profile is determined by the ability of the two-hole final state to delocalise: 4d2 → 4d1 + 4d1 , and as we have seen this is governed by the ratio U(4d2 :LSJ)/W where the LSJ refer to the final state term structure of these transitions. The Pd conduction band density of states will have a strong influence on the Auger profile through its determination of the band-width W, the relationship between the U(LSJ) and the Hilbert transform, H(E), of the Pd density of states and, more subtly, through the values of U(4d2 :LSJ) since the reference energy, or zero, of these quantities is the centre of gravity of the Pd d band. This is the reason why the very different Pd density of states in CuPd and AgPd alloys give rise to such different profiles for the Pd M4,5 N4,5 N4,5 Auger transitions. Davies and Weightman [178] showed that a calculation of the Pd M4,5 N4,5 N4,5 Auger profile using the Pd density of states given by the SCF-KKR-CPA calculations did not give good agreement with the experimental Auger profile and that an arbitrary reduction of the Pd density of states at the bottom of the Pd d band significantly improved the agreement. It was later shown that the Pd density of states given by a model calculation in which the local lattice expansion of 2% measured by EXAFS was included in a Clogston–Wolf treatment of the electronic structure of CuPd significantly reduced the Pd density of states at the bottom of the d band and gave rise a much improved Auger profile [3]. These Auger studies support the conclusion from theoretical work [189–191] that the Pd density of states is significantly influenced by local lattice relaxation in CuPd alloys and contradict Nahm et al.’s [72,75] contention that local lattice effects have very little influence on the Pd density of states. Nahm et al. [75] have sought to discredit the Auger analysis by arguing that Auger profiles will be influenced by transition matrix elements effects similar to those they invoked to account for the discrepancy between the SCF-KKR-CPA results and the Cooper minimum results. In particular they appealed to a study of the L3 VV Auger spectra of V–C where Auger matrix elements are the dominant influence on the Auger profile [192]. Unfortunately this argument is based on a misunderstanding of the Cini–Sawatzky model, a misunderstanding that we shared in our earlier study [178]. When U(4d2 :LSJ) > W, a situation that prevails for almost all the LSJ components of the Pd M4,5 N4,5 N4,5 Auger transitions in both CuPd and AgPd alloys, then a quasi-atomic spectrum is obtained in which the intensity of each LSJ component of the multiplet structure is given by an atomic transition rate. The influence of the Pd density of states is restricted to its determination of W, the role of its Hilbert transform, H(E), in the Cini–Sawatzky formalism and its influence on the U(4d2 :LSJ). There is the possibility that the 4d2 state can hop away from the original site without delocalising but, as shown by Sawatzky and Lenselink [91], this only introduces a slight broadening of the quasi-atomic component. The L3 VV Auger profile of V–C referred to by Nahm et al. [75] corresponds to a system with an incompletely filled band and with low values of U(LSJ) so the quasi-atomic effects captured by CS theory are expected to be small. The Auger profile of this system is an example of the applicability of final state rule, where Auger matrix elements involving band states, which vary with energy for the V–C system, are indeed important, as described above. However while the successful calculations of the profiles of the Pd M4,5 N4,5 N4,5 Auger transitions of CuPd alloys described above is strong evidence in support of the view that lattice relaxations

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Fig. 7. Calculated (solid line) and experimental (dots) Pd M4,5 N4,5 N4,5 Auger spectrum of a Cu rich CuPd alloy. The dashed line shows the background of scattered electrons. Reprinted (Fig. 5) with permission from [193]. Copyright (1998) by the American Physical Society.

cause a significant reduction of the Pd density of states at the bottom of the d band the calculations do depend on a parameterised model, albeit one in which the parameterisation is determined by the measured lattice expansion, and as such are not definitive. What is needed is a first principles calculation of the local Pd density of states in which the measured local lattice expansion is included in the atomic structure of the alloy. This has been carried out by Kucherenko et al. [193] within the linear muffin-tin orbital method using the supercell approach. They find that the local electronic structure on Pd sites is sensitive to local relaxation and that the measured expansion of the lattice obtained from EXAFS reduces the intensity at bottom of Pd d band in a way that is consistent with the results deduced from the Cooper minimum experiment and which also gives rise to an Auger profile in excellent agreement with experiment (Fig. 7). One final point is to note that while Cini–Sawatzky theory has been remarkably successful in explaining the Auger profiles observed from disordered alloys there are, as mentioned above, three outstanding issues which must be addressed before we have a complete understanding of the electron spectroscopy and electronic structure of CuPd and similar alloys. Firstly it is necessary to allow for the effect of off-site interactions [103] and secondly it is necessary to treat correlation and disorder on an equal footing both at the model Hamiltonian and ab initio levels [109]. Finally the theory needs to be extended to include incompletely filled d bands [92]. 7. Conclusions We have shown that experimental information derived from a variety of electron spectroscopy techniques has provided considerable insight into the electronic structure of disordered alloys. The results of such studies have often provided the driving force for improvements in both first principles theory and empirical mod-

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