Ab initio analysis of electron energy loss spectra for complex oxides

Ultramicroscopy 80 (1999) 145}151

Ab initio analysis of electron energy loss spectra for complex oxides S. KoK stlmeier , C. ElsaK sser *, B. Meyer
Max-Planck-Institut fu( r Metallforschung, Seestrasse 92, D-70174 Stuttgart, Germany
Max-Planck-Institut fu( r Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Received 19 January 1999; received in revised form 4 May 1999

Abstract The local electronic structures in ionic materials are analysed in terms of the site- and angular-momentum-projected densities of states, which are obtained from self-consistent ab initio band-structure calculations. The unoccupied conduction-band states are compared to the energy-loss near-edge spectra (ELNES), which are measured by analytical transmission electron microscopy with high spatial resolution. A novel tool for the interpretation of this ELNES data is presented: The projection of the calculated density of states onto an optimised atomic-orbital basis set allows a neighbourhood-sensitive electron-distribution analysis based on crystal orbital overlap populations. This method is applied to spinel, MgAl O , both in its regular and in its inverse structure. An analysis of the O}K-edge gives evidence of   a distinction between the in#uences of tetrahedrally coordinated and octahedrally coordinated cations on the near-edge structure.  1999 Elsevier Science B.V. All rights reserved. PACS: 71.15.!m; 71.15.Ap; 71.15.Hx; 71.15.Mb; 71.20.!b; 71.20.Ps; 82.80.!d; 82.80.Pv Keywords: Electron energy loss spectroscopy; Ceramic materials; Bonding and coordination; Local density functional theory

1. Introduction Recently, the analysis of data from electron energy loss spectroscopy (EELS) on the basis of electronic structure calculations has excited widespread interest [1}4]. The fundamental aspects of the theoretical treatment have been established for a long time [1,5}8], initially for the description of the corresponding X-ray excitation processes by

* Corresponding author. Tel.: #49-711-2095351; fax: #49711-2095320. E-mail address: [email protected] (C. ElsaK sser)

scattering theory [9,10]. With an increasing computer power the use of various ab initio methods has also become feasible, correlating unoccupied energy bands or cluster electronic levels to the near-edge structures (ELNES) visible in the EELS spectrum. Fermi's golden rule is applied which yields a proportionality between the measured intensity I(E) and the product of the square of the calculated transition matrix element "M(E)" for a single electron excitation and the density of the unoccupied electron states N (E) (see, e.g., Ref. [6]). S This relation is based on the assumptions that the EELS process can be described by a one-electron excitation and that this excitation happens within an in"nitesimally short time interval. For

0304-3991/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 9 9 ) 0 0 1 0 3 - 5

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near-parallel scattering the dipole selection rules apply as well as in optical spectroscopy. The density of states N (E) is obtained by band S structure calculations. In order to obey the dipole selection criteria the appropriate angular-momentum component of N (E) can be obtained by S projection onto spherical harmonics within the atom-centred spheres or by the calculation of the matrix elements M(E). Modern band structure methods (all-electron as well as pseudopotential techniques, see, e.g. Refs. [11,12]) have been demonstrated to give EELS K-edges in satisfactory agreement with the experimental data [1}3]. The treatment of more complex excitation processes, which give rise to ¸ -edges for instance, has been  discussed in detail recently [13]. Evidence was given for a `"ngerprinting schemea for the long-range oxygen arrangement of binary and ternary oxide crystals based on an analysis of the EELS signals in the oxygen K-edge by comparison with the site and angular-momentum projection of N (E) for l"1 [13]. In this work it will be S shown that the short-range interactions, i.e. the interactions of the oxygen ion with the directly neighbouring cations, leads to characteristic "ngerprints, as well. Two materials ideally suited for this investigation are normal and inverse spinel MgAl O . These crystals provide two di!erent   divalent and trivalent cations in both tetrahedral and octahedral oxygen coordination, but nevertheless all the oxygen sites in each structure are crystallographically equivalent. Hence, all features in the O}K-edge arise from the same oxygen ion and no complicated overlay of edges due to di!erent local chemical environment has to be considered. At "rst glance this advantage has a drawback: Oxygen is coordinated by a distorted tetrahedron of three Al ions and one Mg ion, which gives rise to a complicated "ne structure. However, the projection of N (E) onto the angular momentum function S for l"1 averages over all the interactions between the oxygen ion and its four nearest neighbours within the projection sphere around the oxygen site in a similar fashion as the experiment does. Therefore, the angular-momentum projection does not yield the in#uence of each di!erent cation neighbours on the O}K-edge separately. To achieve this, N (E) was projected alternatively onto optimised S

atomic orbitals [14]. From this projection crystal orbital overlap populations between the pairs Al}O and Mg}O were calculated, which allow for an analysis of the spectrum explicitly in terms of the two-centre Al}O or Mg}O interactions.

2. Computational method The ab initio electronic structure calculations are based on the local density functional theory (see, e.g. Refs. [11,12]). The Kohn}Sham e!ective oneelectron equations are solved self-consistently for the valence electron states represented by a mixed basis set of plane waves (cuto! energy: 20 Ry) and additional local orbitals [15,16] for p states of O. The core-valence interactions are represented by norm-conserving ionic pseudopotentials. The site- and angular-momentum-resolved local DOS (L-DOS) are obtained from the total DOS by projection onto atom-centred spherical harmonics > (h, ) within the projection spheres [17]. No JK mu$n-tin spheres for the construction of basis functions (or eventually of appropriate one-electron potentials) such as in LMTO or LAPW calculations are required in the present band structure approach. Therefore the choice of the sphere radii for the L-DOS projection was given special attention. For oxygen a radius of 1.59 As was employed, which ensures touching spheres on the slightly distorted fcc oxygen sublattice. With radii of 0.76 As the spheres around the cations are slightly larger than the available space for non-overlapping spheres centered at the interstitial tetrahedral or octahedral sites of the oxygen sublattice. By this choice the sum of the electronic charges contained in all the projection spheres together is equal to the total charge of the unit cell (64 electrons for 2 formula units of spinel). For the determination of crystal orbital overlap populations [18] the one-electron wave functions were projected onto a basis set of modi"ed local pseudo-atomic s- and p-type orbitals centred at each site A [14] u ( r )"R (r)iJK ( r( ), LJ JK LJK where n, m, and l are quantum numbers, R denotes the radial and K the angle-dependent part. For the

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radial part R (r) the radial pseudo-atomic wave LJ functions u (r) were spatially con"ned by a cuto! LJ function into a sphere with radius r [16]:  R (r)"Cru (r)j (1!e\ALJALJ\P) for r(r . LJ LJ LJ  The parameters j and c can be optimised to LJ LJ yield a minimum spillage [14,17]. The spillage is the relative loss of electron charge due to the incompleteness of the pseudo-atomic-orbital projection. The constant C guarantees the normalisation of the resulting contracted function. In order to analyse only the nearest-neighbour bonding region the same values were chosen for r as used for the angular-momentum projection.  This procedure leads to an electron spillage of 2.6% per electron in normal and 2.3% in inverse spinel. If the orbitals were optimised with respect to their spatial extension, a spillage of only 0.1% per electron was obtained for normal spinel with orbitals optimised independently for bulk MgO and aAl O crystals.   3. Model systems The smallest possible crystallographic unit cell of spinel, which is face-centred cubic and contains 14 atoms (as reported for normal spinel in Ref. [19]), was used in the calculations and repeated by threedimensional periodic boundary conditions. In this compound the oxygen ions form a distorted cubic close packed sublattice. One half of the octahedral sites in this sublattice is occupied by Al ions (Al(oct)), one-eighth of the tetrahedral sites by Mg ions (Mg(tet)), both in a highly ordered fashion in which the sublattice of the two cations together corresponds to the structure of the cubic Laves phase (C15). Thus the O ions are coordinated by distorted tetrahedra of one Mg and three Al ions with a local C symmetry around the Mg}O axis. T In normal spinel there exists only one crystallographic position for each of the three constituing atoms. This cation arrangement in the oxygen sublattice is predominantly found in natural spinel, whereas synthetic spinel shows some degree of inversion, where the two cation species change sites. In the extreme case of the completely inverse structure, all

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Fig. 1. Comparison of measured O}K-edge in spinel (courtesy of H. MuK llejans) and calculated p-L-DOS at the O site in normal (n) and inverse (i) spinel.

Mg ions are situated in octahedral sites and one half of the Al ions is transferred to tetrahedral sites. If one interchanges two Mg and two Al ions in this smallest unit cell, one obtains a highly ordered inverse spinel structure. All Mg ions are located at equivalent octahedral sites (Mg(oct)), whereas two crystallographically di!erent positions are occupied by Al, an octahedral coordinated Al(oct) site and a tetrahedrally coordinated one, Al(tet). The analysis of the corresponding Al-edges in the EELS is complicated, because it must account for the superposition of the signals which originate from both the Al(oct) and the Al(tet) sites. The O ions, on the other hand, occupy only one crystallographic position, where they are surrounded by distorted tetrahedra of one Mg and three Al ions at a lower local C symmetry. Q The experimental O}K-edge spectrum of natural spinel, that is given in Fig. 1 for comparison to our calculated results, was recorded by H. MuK llejans on a dedicated scanning transmission electron microscope installed at the MPI fuK r Metallforschung in Stuttgart, a VG HB501 UX STEM with 100 keV acceleration voltage and equipped with a Gatan 666 PEELS detector. The spectra were obtained with an energy dispersion of 0.1 eV/channel and an analytical form of the background was subtracted

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(for more details on the experimental technique see e.g. Ref. [20]).

4. Calculation of O}K-edges of normal and inverse spinel An attempt was reported in the literature [21] to determine the degree of inversion by an analysis of the experimentally recorded cation ¸ -edges in  the EELS. A noticeable shoulder occurs at the low energy-loss side of the "rst signal of the Mg-L  and Al-L -edges with increasing degree of inver sion. For several normal and partially inverse spinel samples the cation L -edges were decom posed into two Gaussian peaks, and the intensity ratio was interpolated linearly with the degree of inversion. As a complication to this approach the Mg-L - and the Al-L -edges overlay and there  fore a model background was subtracted from the Al-L -edge instead of the pre-edge intensity.  Within the assumptions made in that study, however, only qualitative results could be obtained, and the authors of Ref. [21] suggest that theoretical investigations could be helpful for a deeper understanding. Considering the complicated nature of the overlapping metal L -edges, however, the  simpler O}K-edge was chosen for the present study, for which changes were reported as well [21]. Earlier EELS observations of numerous oxide crystals exhibit a signi"cant splitting of the "rst signal of the O}K-edge in cases where the oxygen is tetrahedrally surrounded by cations [23,26]. This feature, which was proposed to be a characteristic "ngerprint for the tetrahedral environment of oxygen, is present in the O}K-edge of spinel as a shoulder on the low-loss side of the "rst peak. The cation tetrahedron coordinating the oxygen consists of Mg(tet) and Al(oct) in normal spinel, and it undergoes a pronounced change to the more complex coordination by Mg(oct), Al(oct), and Al(tet) in the inverse structure. Hence, the oxygen sites act as the nearest-neighbour spectators of the cation inversion, and the O}K-edge contains all the necessary information about the degree of inversion, in the speci"c case of MgAl O without any further ef  fects due to hybridisation of the O-2p-states with

metal d-states as, for instance, in titanates [24]. However, the energy resolution necessary for a unique di!erentiation between inverse and normal structure can not be obtained under the currently available experimental ELNES conditions. Therefore, an experimentally recorded O}K-edge displayed in Fig. 1 does not directly allow for a quantitative determination of the degree of inversion. Additionally, a reference EELS spectrum of the completely inverse spinel has not been reported in the literature to the best of our knowledge. According to our calculation the inverse MgAl O   structure is less stable than the normal one by 1.12 eV per formula unit. To complement these experiments, ab initio calculations were carried out on the two idealised spinel structures described above. Fig. 1 displays the local density of unoccupied states (L-DOS) projected on the angular momentum l"1 for both compounds, (n) for normal and (i) for inverse spinel. The calculated data are convoluted with a Gaussian function of 1 eV full-width at half-maximum which accounts for the energy broadening of the electron beam inside the microscope. The zero point of the energy scale is arbitrarily chosen as the energy of the last occupied band in the calculated L-DOS. The band gaps were calculated to amount to 4.9 eV (inverse) and 6.6 eV (normal), underestimating the experimental value of 7.8 eV from optical re#ectivity measurements [25]. (This theoretical underestimation of the optical energy gap is a well established de"ciency of the local-density approximation in the density functional theory.) The experimental EELS data were aligned at the energy of the global intensity maximum (at 14 eV). Within the "rst 40 eV the experimental spectrum exhibits three distinct maxima which occur at 14, 21, and 33 eV, respectively, of the chosen energy scale. Additionally, two shoulders (sh) appear at approximately 11 and 28 eV. The features in the calculated L-DOS are sharper and located at 10 eV (sh), 14, 20, 26.5, and 33 eV in normal spinel and at 6.5, 11.5 eV (sh), 14, 21, 25.5, 31 eV (sh), and 33 eV in the inverse case. In the comparison of the two calculated spectra it is obvious that the splitting pattern of the O p-LDOS is more complicated for the inverse structure. The "ve features present in the p-L-DOS of normal

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spinel are obtained in the inverse case, as well, with an average deviation in peak positions of 0.3 eV. The major di!erences are an additional peak at the edge onset and another shoulder at 31 eV for the inverse spinel which re#ect the reduced symmetry and the higher anisotropy of the crystal "eld around the oxygen centre. The absence of these features in the experimental spectrum leads to the conclusion that in the studied sample the cation ordering is close to normal. The results also imply that the increase of the "rst shoulder in the synthetic spinel as observed by Bruley et al. [21] can in fact be attributed to inversion. For a further analysis the calculated one-electron wave functions were projected [14] onto modi"ed local pseudo-atomic orbitals at site A using the same cuto! radius r as for the L-DOS projection  spheres. From these orbital projections the conventional population analyses can be carried out. A Mulliken analysis [22], for instance, yields anionic picture for both spinels with partial charges of q(O(2s))"q(O(2p))"1.9 electrons per orbital and no signi"cant contributions from the cations. More important for the understanding of the ELNES features is the calculation of the crystal orbital overlap populations (COOP) between orbitals u and uY at neighbouring sites A and A. This JK JYKY quantity, denoted as COOP (Al,Al,E), is site-resolved and energy-resolved such as the L-DOS, and it additionally provides bond-resolved information about the interaction of speci"c atom pairs. In the present study the COOP (Al,Al,E) is employed as an analytic tool to display in which energy range the interaction between the sites A and A contributes to the total or a local DOS. The sign of the COOP(Al,Al,E) can also be interpreted as an indicator, whether this interaction is bonding (COOP'0) or antibonding (COOP(0) [18]. To the best of our knowledge this COOP scheme using pseudo-atomic orbitals with optimised slopes has been applied to the interpretation of EELS data for the "rst time in the present study. Before the results of this novel analytical tool are given, it should be noted, however, that the COOP [18] can not rigorously be related to experimentally observable quantities. Therefore, the partitioning of the electron density into COOP involves some arbitrariness concerning the choice of the basis functions,

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Fig. 2. O p-L-DOS and COOP involving the O(p) orbital for normal (left panels) and inverse spinel (right panels). Depicted are the O p-L-DOS (1st row), and the COOP at the tetrahedral (2nd row) and the octahedral sites (3rd row). Broken lines are used to distinguish Mg from Al sites. The energy is given in eV, the DOS in eV\, and the COOP plots within equal intensity windows of 1.2 eV\. For purpose of clarity the COOP curves are linearly shifted with respect to each other on the same arbitrary inverse energy scale. Their zero levels are indicated on the right hand side of the panels.

such as the focus on only the nearest-neighbour interaction in the present study. Fig. 2 contains all possible interactions between the O(p) and the metal M(s) and M(p) orbitals in normal spinel (left panels) and inverse spinel (right panels), together with the O p-L-DOS, both the occupied (E(0 eV) and the unoccupied part (E'0 eV). Due to the very ionic nature the COOP do not display much intensity in the occupied part of the spectrum which re#ects the charge localised mainly on O. In the unoccupied part the delocalised, unbound states have a much higher probability between the ionic centres, hence, the higher values for the COOP. For energies above 25 eV again only small values for COOP(Ms,Op,E) and COOP(Mp,Op,E) are obtained. This may be interpreted as an indication that long-range interactions start to dominate that part of the spectrum, which

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were excluded in the present analysis by the choice of r .  First, only the occupied states will be discussed. In normal spinel the interaction between O(p) and both Mg(s) and Mg(p) is mainly bonding, because the corresponding COOP(Mgs,Op,E) and COOP(Mgp,Op,E) are positive. The predominantly negative values for COOP(Als,Op,E) and COOP(Alp,Op,E) indicate that the Al(s)-O(p)- and Al(p)}O(p)-interaction is antibonding. An analysis of the inverse spinel exhibits the interesting e!ect, that the bonding or antibonding character is site dependent. For the inverse structure the negative COOP(Mgs,Op,E) and COOP(Mgp,Op,E) indicate antibonding occupied states, whereas the tetrahedrally coordinated Al ions are bonding to O. As a consequence of the cation reordering in inverse spinel also the octahedrally coordinated Al ions are now at bonding sites. Only minor qualitative di!erences between the COOP(Ms,Op,E) and COOP(Mp,Op,E) are obtained, i.e. both metal M(s) and M(p) orbitals follow the same trends. In the energy range 0 eV(E(25 eV the COOP calculated for the unoccupied states exhibit approximately the reverse characteristics of the ones of the occupied states in all the "ve cases. For the shoulder at the O}K-edge onset in normal spinel the predominant interaction of O(p) is with the Mg(p) orbital and, to a smaller degree, also with Mg(s), i.e. with the orbitals of the metal in the tetrahedral position. The values for COOP(Als,Op,E) and COOP(Alp,Op,E) are comparatively smaller and exhibit signi"cant contributions only around the maximum intensity. A similar trend is observed in the COOP calculated for the inverse spinel: Here, Mg is octahedrally coordinated and thus does not contribute to the interactions at the O}K-edge onset. On the contrary, the COOP(Mgs,Op,E) and COOP(Mgp,Op,E) show a striking similarity to the corresponding values for an Al site in the normal spinel with two major features in COOP(Mgs,Op,E) and one in COOP(Mgp,Op,E). Again, the Al ion in the octahedral position is involved only around the intensity maximum, although in an antibonding fashion. The main focus is drawn to the COOP including the tetrahedrally coordinated Al ion in the lowest panel. In

this case, both the COOP(Als,Op,E) and the COOP(Alp,Op,E) are qualitatively similar to the values typical for the tetrahedrally coordinated Mg centres of the normal structure. This indicates that the peak at the edge onset can be classi"ed due to the tetrahedral coordination at the oxygen site.

5. Summary A novel environment-sensitive electron-distribution analysis (with respect to sites, bonds, and energies) for data from self-consistent ab initio band-structure calculations has been presented and applied to the assignment of features in the ELNES of oxide materials. For the present example, normal and inverse spinel, the following conclusions can be given: E The "rst peak in the O}K-edge of these cubic close-packed oxide ceramics is due to the interaction between the O(p)-orbitals and the s- and p-orbitals of the cation in the tetrahedrally coordinated site. In the spinel case the common notion is justi"ed that the peak at the edge onset originates from tetrahedral nearest-neighbour coordination. E Peaks at higher energy losses with respect to the edge onset show no contribution from the nearest neighbour interactions. This is an indication that long range interactions dominate that part of the spectrum. E The analysis of the occupied states underlines the highly ionic character of both the normal and inverse spinel. For the remaining interatomic interactions between occupied levels, a site-dependence is observed: bonding occurs between O and the tetrahedrally coordinated cations and anti-bonding is obtained between O and the cations in the octahedral sites. The presented procedure has been shown to work out even delicate details of the electronic structure related to changes in the local chemical environment. The favourable comparison with the experimental EELS data for the unoccupied electronic states in the present study gives support to the analysis of bonding via calculations of COOP for

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occupied states. Current investigations therefore, include an application of this new tool to the analysis of the bonding at grain boundaries and heterophase interfaces.

Acknowledgements This work is supported by the Volkswagen Stiftung, Germany (project No. VW I/70502).

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