Journal of Physics and Chemistry of Solids 64 (2003) 495–505 www.elsevier.com/locate/jpcs The electronic structure of Be and BeO: benchmark EMS measurements and LCAO calculations B. Soule de Basa, H.E. Dorsettb, M.J. Fordc,* ´ a School of Chemistry, Physics and Earth Science, Flinders University of South Australia, G.P.O. Box 2100, Adelaide, SA 5001, Australia b Defence Science and Technology Organisation, P.O. Box 44, Pyrmont, NSW 2009, Australia c Institute of Nanoscale Technology, University of Technology Sydney, P.O. Box 123, Broadway, NSW 2007, Australia Received 19 April 2002; accepted 26 July 2002Abstract The electronic band structures of Be and BeO have been measured by transmission electron momentum spectroscopy (EMS).The low atomic number of beryllium and the use of ultrathin solid ﬁlms in these experiments reduce the probability of electronmultiple scattering within the sample, resulting in very clean ‘benchmark’ measurements for the EMS technique. Experimentaldata are compared to tight-binding (LCAO) electronic structure calculations using Hartree – Fock , and local density (LDA-VWN), gradient corrected (PBE) and hybrid (PBE0) density functional theory. Overall, DFT calculations reproduce the EMSdata for metallic Be reasonably well. PBE predictions for the valence bandwidth of Be are in excellent agreement with EMSdata, provided the calculations employ a large basis set augmented with diffuse functions. For BeO, PBE calculations using amoderately sized basis set are in reasonable agreement with experiment, slightly underestimating the valence bandgap andoverestimating the O(2s) and O(2p) bandwidths. The calculations also underestimate the EMS intensity of the O(2p) bandaround the G-point. Simulation of the effects of multiple scattering in the calculated oxide bandstructures do not explain thesesystematic differences.Crown Copyright q 2002 Published by Elsevier Science Ltd. All rights reserved.Keywords: A. Metals; A. Oxides1. Introduction scattering of the electrons within the sample [4,5]. However, the signal-to-noise ratio is vastly improved when the solid In the same way that electron momentum spectroscopy ˚ samples are thin (, 100 A), or when they are composed of(EMS) has previously been used to probe the electronic light elements. In this sense, beryllium and beryllium oxidestructure of atoms and molecules in the gas phase [1,2], at provide ideal standards with which to test the limits of EMSFlinders we use the technique to provide measurements of on solid samples, and the two systems are complementarythe electronic band structure of the condensed phase . examples of metallic and insulating solids. Beryllium has aEMS techniques have been reﬁned to the point that it is now low atomic number (Z ¼ 4), and ultrathin ﬁlms of Be orpossible to map electron momentum distributions with a BeO are readily grown on carbon substrates using standardresolution approaching ‘chemical accuracies’ (, 0.2 eV), deposition techniques. The consequent reduction in multipleallowing a quantitative comparison between measurements scattering yields a relatively clean signal in which plasmonand theoretical calculations. excitation and small angle elastic scattering effects are Transmission EMS on solid samples is generally readily distinguishable from ‘true’ sampling of the bulkdifﬁcult, since measurements are subject to multiple electronic band structure. The low atomic number of beryllium also facilitates * Corresponding author. Tel.: þ61-2-9514-7956; fax: þ 61-2- theoretical studies of these systems. Beryllium has been9514-7553. the subject of solid state calculations for nearly 60 years, yet E-mail address: email@example.com (M.J. Ford). still presents a unique challenge to modelling efforts.0022-3697/03/$ - see front matter Crown Copyright q 2002 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 2 - 3 6 9 7 ( 0 2 ) 0 0 3 5 6 - 6
496 ´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505Although solid Be is nominally metallic, the inter-crystal- experimental measurements to the LCAO calculations, andline bonding has directional ‘covalent’ properties and the discuss in more detail some of the conclusions we draw fromsolid displays a number of departures from free-electron- these results.like metallic behaviour [6,7]. Also, correlation effectsbetween the two electrons in outer valence shell of Bemust be taken into account in any realistic model [8,9]. 2. Experimental methodsSimilarly, the oxide is often characterised as ionic, yet thereare several calculations that predict interstitial charge anddirectional bonding [10,11]. A number of detailed accounts of EMS can be found in Calculations for Be and BeO have traditionally been the literature, here we give an overview of the salient pointscompared with optical, X-ray and photoemission measure- of the experiment [1,2,4]. EMS is an electron impactments [12 –18]. Generally, these techniques map certain technique, which utilises ionisation of the target to probeaspects of the band structure such as special-point energies electronic structure. In the Flinders spectrometer [21,22], anand densities of states, or require considerable theoretical incident electron beam of nominally 20.8 keV energyinput for their interpretation. The band dispersions for Be impacts on an ultrathin, self-supporting solid target. Twohave previously been mapped by angle resolved photoemis- energy and angle selective electron analysers are positionedsion (ARPES) , however, a full interpretation of the on the opposite side of this target to the incident beam. Onepeak intensities observed in these measurements is analyser is set to detect electrons leaving the impact atcomplicated by the electron and photon interactions nominally 19.6 keV and 13.68 relative to the incident beam,involved. To our knowledge, ARPES measurements for and the other at nominally 1.2 keV and 768. A schematicBeO have not previously been reported. diagram of the scattering geometry is shown in Fig. 1(a). The emphasis of the present work is to present our EMS Electrons detected in the analysers can originate from ameasurements of the valence band structures of Be and BeO variety of scattering events—in EMS we are interested into demonstrate the utility of the technique and provide a set only a small fraction of these: single ionisation of the target.of benchmark experimental data. EMS can measure directly To discriminate these ionisation events from unwantedthe band dispersions and intensities, and provides events, a time coincidence detection method is used, whereinformation complementary to photoemission and other the time difference between pairs of electrons detected at theestablished techniques. In addition, the metal and its oxide two analysers is measured. Two electrons which originateare complementary types of solids representing more from the same ionisation event will arrive at the twoitinerant or more localised electronic systems, respectively. analysers at essentially the same time, i.e. in coincidence.Whether a particular theoretical method is ‘universal’ can Pairs of electrons from other events give a randomthen be tested to a reasonable degree by application to both distribution of time differences.these solids. We compare our measurements to tight-binding The scattering geometry of the spectrometer is chosen so(LCAO) calculations using the CRYSTAL98 suite of that the ionisation event can be described as a directprograms . The calculations have been spherically knockout of the target electron by the incident electron—theaveraged to account for the structure of the target, and so-called (e,2e) process —with momentum transferredmultiple scattering effects included for both systems using a only to the target electron. The ion core is a spectator in theMonte Carlo procedure. For both Be and BeO, we ﬁnd process. Hence, we know the energies and momenta of thereasonable agreement between EMS measurements and incident and two outgoing electrons and can thereforecalculations using density functional methods (DFT) coupled determine the energy and momentum of the target electronwith moderately sized basis sets. There is, however, a the instant before it was ejected from the target. Thesystematic overestimation in the predicted bandwidths for analysers detect electrons over a small window of energiesBeO even after the inclusion of spherical averaging. For and range of angles. By measuring the number ofmetallic beryllium, we perform calculations with two coincidence events as a function of the energies and anglesdifferent sizes of basis sets, and ﬁnd that bandstructures of the two electrons we build up a probability map of thepredicted by the larger basis are in better agreement with the bound target electrons having a particular binding energyexperimental results. The predicted band intensities for Be and momentum. In other words, EMS provides a directshow signiﬁcant deviations from the experimental data which measurement of the band dispersion and density of occupiedcan be explained in terms of many-body effects. For BeO we states for the target solid.observe considerable intensity in the O(2p) band around the Since this is a transmission measurement, the target hasG-point whereas the predicted intensity is small, a difference to be extremely thin in order to minimise multiple scatteringwhich is not readily explained by multiple scattering. of the incident or outgoing electrons. In practice targets In the following sections, we ﬁrst give speciﬁcs of must be self-supporting and no thicker than about 10 nm.the EMS measurement, the sample preparation and Targets are easily prepared by standard evaporationthe theoretical calculations in this study. We next present techniques; Be is evaporated onto an amorphous carbonthe data analysis, including a quantitative comparison of our ﬁlm nominally 3 nm thick to form an overlayer of
´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505 497Fig. 1. (a) Schematic of the spectrometer scattering geometry, shaded areas represent angular acceptance of analysers. (b) Target orientationwith respect to incident and outgoing electrons, shaded region represents portion of target contributing to (e,2e) signal.approximately the same thickness. Evaporation is carried Vosko– Wilk – Nusair correlation , and GGA resultsout in situ under UHV conditions, and the base pressure of were obtained with Perdew – Burke – Ernzerhof (PBE)the evaporation chamber is about 10210 T rising to typically exchange and correlation . The hybrid method used1028 T during evaporation. The oxide samples were was PBE0, developed by Adamo and Barone [27,28].prepared by heating Be targets to about 600 8C in an oxygen Calculations were performed with all electron basis setsatmosphere of 3 £ 1027 T for about 20 min. Overlayer speciﬁcally optimised for Be and BeO. For metallicthickness was monitored using a quartz crystal microba- beryllium, two basis sets were used: a small 5-111G basislance. The relatively small escape depth of the slower developed by Dovesi et al. , and an extended 5-1111-1.2 keV ejected electron means that the coincidence signal ¨ (3p)-(2d) basis—‘vtz1’—developed by Gruneich andcomes predominantly from about the outermost 2 nm of the Hess . The vtz1 basis contains diffuse s functionstarget. This is sufﬁciently depth sensitive that the am-C speciﬁcally optimised to yield energies within 1 mH of thesubstrate does not contribute to the measured band structure HF limit, yet avoid numerical errors due to linearyet not so surface sensitive that reaction of the surface with dependencies in periodic calculations. For BeO, the valenceresidual gases present in the vacuum during the electronic structure is comprised mainly of O(2s) and O(2p)measurement becomes a problem. Data are collected overa 2 – 3 day period in order to build up sufﬁcient statistics. The quality of the prepared targets can be characterisedusing Auger electron spectroscopy as shown in Fig. 2(a) and(b). In Fig. 2(a) the Be KVV line is clearly present togetherwith a small peak at 270 eV attributable to the carbon KLLline. In Fig. 2(b) the Auger spectrum shows a clear shift ofthe Be line together with the O KVV line at 512 eV indictingformation of the oxide.3. Computational methods3.1. Ab initio LCAO calculations Theoretical band structures for Be and BeO were derivedfrom self-consistent Hartree – Fock (HF) or density-func-tional tight-binding calculations using CRYSTAL98 .DFT calculations included a local density approximation(LDA), a generalised gradient approximation (GGA), and ahybrid method incorporating HF exchange. LDA results Fig. 2. Differential Auger spectra for (a) the metallic Be target andwere obtained with Dirac – Slater exchange  and (b) the BeO target.
498 ´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505orbitals, so only moderately sized basis sets were used in averaging of the bandstructure calculation was performedthe calculations, namely 5-11G for Be2þ and 8-411-(1d)G via Monte Carlo sampling of the irreducible wedge of thefor O22 developed by Lichanot et al. . Brillioun zone (IBZ) of Be or BeO. The Monte Carlo To assess the performance of the two Be basis sets, procedure produces 25 equally spaced crystal directionsminimum-energy structures and cohesive energies were within the IBZ, and the band dispersion and electroncalculated for each basis using HF and DFT methods. momentum density (EMD) is then calculated usingResults are listed in Table 1. Calculated values for the HF CRYSTAL98 along these 25 directions. The EMD is foldedresults are slightly different than those reported by Dovesi into the corresponding dispersion curve and the 25 results ¨et al. , and Gruneich and Hess , with the variance summed together. The spherical averages are then con-(,1%) likely due to different SCF convergence properties voluted with Gaussian functions representing the exper-and tolerances used in the respective calculations. The vtz1 imental resolutions in momentum and energy. The resultingbasis gave better predictions for the experimental lattice energy and momentum resolved density can then beparameters  than 5-111G for HF and DFT, albeit with a compared directly to the output of the EMS spectrometer.nearly 10-fold increase in computational effort. However, Despite the thin targets and relatively low atomiccohesive energies closer to the experimental value of number of the targets, the experimental data are still76.6 kcal/mol  were obtained with the smaller basis inﬂuenced by multiple scattering. The incident or outgoingusing PBE and PBE0 DFT. electrons can undergo, in addition to the ionisation event, A comparison of the calculated LDA band dispersions other elastic or inelastic scattering processes, giving rise to aand electron momentum densities for metallic Be using the background in the EMS data. In order to account for this in5-111G and vtz1 basis sets are shown in Fig. 3. Here and the calculated data we have developed a Monte Carloelsewhere in this paper, electronic structures have been procedure to simulate multiple scattering that can be appliedcalculated at the experimental lattice parameters. The to both the (monatomic) metallic and (diatomic) ionicexperimental band dispersion derived from the EMS data samples. The input to the code is the spherically averagedis also shown and will be discussed in Section 4. EMS LCAO calculation. The Monte Carlo procedure is used todispersion is a spherical average over all crystal directions determine whether and how many multiple scattering eventsand experimental data is the same in each panel of Fig. 3. It the electrons undergo and modiﬁes the LCAO energy–is clear in Fig. 3 that the use of the more computationally momentum density accordingly. Probabilities are assignedexpensive vtz1 basis set yields a slight improvement in the to the various processes—namely bulk and surface plasmonpredicted electronic structure, with the 5-111G basis set excitation, valence electron excitation, and elastic scatter-giving a band dispersion 5% larger than the vtz1 basis. ing—based upon their mean-free path as calculated from input parameters.3.2. Spherical averaging and multiple scattering The code used in the present work is based upon that of Vos and Bottema  but contains an improved description The experimental targets prepared by the above methods of the scattering processes and a simple model to simulateare expected to have a polycrystalline structure, therefore systems containing two atom types. For BeO, a calculatedthe measured band structures represent a spherical average Rutherford scattering cross-section from the O22 and Be2þover all crystal directions. In order to directly compare the ions has been used to simulate the elastic contribution totheoretical models to EMS measurements, a spherical ˚ multiple scattering. A cut-off of 0.5 A has been applied toTable 1Calculated and experimental lattice constants and binding energies. The percent difference between the calculated and experimental values aregiven in parenthesesBasis and method ˚ a (A) ˚ c (A) Emin (a.u.) Eb (kcal/mol/atom)HF/5-111G 2.2312 (22.39) 3.4878 (22.69) 229.2500 45.2LDA/5-111G 2.2016 (23.68) 3.4908 (22.61) 229.1564 94.8PBE/5-111G 2.2080 (23.40) 3.4846 (22.78) 229.4763 80.9PBE0/5-111G 2.1961 (23.92) 3.4694 (23.21) 229.4879 79.7HF/vtz1 2.3141 (þ 1.24) 3.5959 (þ 0.32) 229.3054 56.5LDA/vtz1 2.2268 (22.58) 3.5064 (22.17) 229.2032 103PBE/vtz1 2.2610 (21.08) 3.5622 (20.62) 229.5319 91.6PBE0/vtz1 2.2583 (21.20) 3.5523 (20.89) 229.5419 90.4Experiment 2.2858 ^ 2a 3.5843 ^ 2a – 76.6b a Ref. . b Ref. .
´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505 499Fig. 3. DFT band dispersions and electron momentum densities for Be calculated with the LDA. Experimental (spherically averaged) banddispersion derived from EMS data also shown. Binding energies for the two calculations are relative to the Fermi energy, experimental bindingenergies are normalised to the 5-111G basis set calculation at the band minimum. Momenta are given in atomic units (a.u.).these cross-sections since they tend to inﬁnity as The left-hand panels in Fig. 4(a) and (b) show thethe scattering angle approaches zero. For metallic Be, the spherically averaged LDA calculations with the MonteBorn cross-section for the neutral atom is used, and no cut- Carlo simulation for Be (5-111G basis) and BeO. In the caseoff is required as the cross-section is ﬁnite over all scattering of the Be calculation, the Monte Carlo simulation includesangles. an additional convolution to account for electron – holeFig. 4. Energy– momentum resolved densities. The calculation is the spherically averaged DFT LDA with a Monte Carlo simulation of multiplescattering and lifetime effects for Be. Calculated binding energy has been normalised to the experiment at zero momentum. Intensity is on alinear gray scale.
500 ´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505lifetime effects which are known to be important in the and we believe this is due to the carbon substrate onto whichmetal . The effect of spherical averaging is close to a the Be is evaporated.simple summation over the ﬁve directions shown in Fig. 3. To provide a quantitative analysis of the data, we takeSome of the details within the band structure will tend to be vertical slices through the data in Fig. 4 spanning narrowsmeared out in the process, such as the band gaps or small momentum ranges (typically 0.1 or 0.05 a.u.). The result is adepartures from free-electron behaviour at the Fermi series of binding energy proﬁles at different momentumsurface. Multiple scattering produces a background in the values, and the peaks in these spectra can then be ﬁtted todata which smears intensity along the energy axis for extract the band dispersions, bandwidths and intervalenceinelastic processes, and along the momentum axis for elastic bandgaps. A least squares ﬁt using a combination of a thirdscattering. Small angle elastic scattering is the dominant order polynomial for the background and one or twomultiple scattering effect in the oxide, resulting from a rapid Gaussians for each binding energy peak was used for bothincrease in the Rutherford cross-section at small angles and the Be and BeO experimental data. Two Gaussians werehence small mean-free path for the process. Conversely, in required to account for small shoulders present on the peaks,the metal it is inelastic processes that make a signiﬁcant and in all cases the ﬁt qualities were extremely high. Fittingcontribution producing an overall background and ‘ghost’ the data in this manner allows us to determine the peakbands below the valence band due to valence electron and positions to greater precision than the independent error inplasmon excitations. individual data points, or resolution of the experiment. From the least squares ﬁtting procedure, we estimate that the overall precision of peak positions is ^ 0.15 eV. The spherically averaged calculations have also be ﬁtted in a4. Results and discussion similar manner. We present and discuss the results of this analysis for Be and BeO in Sections 4.1 and 4.2. The right hand panels of Fig. 4(a) and (b) show the 4.1. Bemeasured energy and momentum resolved densities for Beand BeO, respectively. The binding energies in these plots Experimental and calculated binding energy proﬁlesare relative to the vacuum level of the spectrometer; the over 0.1 a.u. momentum intervals are shown in Fig. 5. Asame reference point will be used throughout this paper. smooth background has been subtracted from the EMS andCalculated binding energies in Fig. 4 have been shifted to calculated data using the third order polynomial. Thematch the G-point energy of the valence band from the calculations are spherically averaged LDA/5-111G andexperiment. This accounts for the fact that the calculation is LDA/vtz1, and include a simulation of multiple scatteringperformed relative to the Fermi level. and electron – hole lifetimes through the Monte Carlo As EMS measures a real electron momentum rather than procedure. Comparing calculated results obtained with thecrystal momentum, our band structures are in the extended two different basis sets, it is clear that the extended vtz1zone scheme and are labeled in atomic units (a.u.) of yields slightly different results to the 5-111G basis, as canmomentum rather than the more familiar symmetry points also be seen in the calculated dispersion curves of Fig. 3, andof the IBZ. The overall energy and momentum resolution of holds true for the HF and DFT Hamiltonians also employedour spectrometer is 1 eV and 0.1 a.u., respectively. in this study. The predicted band structures and background simu- There is a reasonable overall agreement between thelations reproduce the general features of the measurement in predicted peak positions and intensities compared with theFig. 4(a) and (b). For Be, the result is a simple parabola experimental data. Agreement in the peak intensitiesindicative of a metallic (free-electron-like) system which requires lifetime effects to be included in the calculation.stands out clearly from the multiple scattering background. The inverse lifetime is largest at the bottom of the bandA faint second parabola can be seen at an additional energy (q ¼ 0.0 a.u.) and decreases up the parabola. The intensityloss of about 19 eV which is due to the excitation of a single within the band is broadened according to the magnitude ofbulk plasmon by the incoming or outgoing electrons. This the inverse lifetime. Hence, the net effect is to distributefeature is reproduced by the Monte Carlo simulation. intensity along the parabola, with the maximum intensity The BeO bandstructure is characterised by two valence pushed away from the band minimum toward largerbands derived from the O(2p) and O(2s) orbitals. As momentum values [19,33].expected for an insulating system, both bands are less To provide a direct comparison among all the theoreticaldispersive than for metallic Be. Once again the EMS signal methods, we list bandwidths in Table 2. These are obtainedis relatively clean and the band structure stands out clearly by ﬁnding the energy difference between the calculatedfrom the multiple scattering background. Small angle elastic Fermi energy and G-point energy in the lowest dispersionscattering is the dominant process which tends to smear curve in Fig. 3. A useful comparison can now be made byintensity within the band structure along the momentum obtaining the experimental bandwidth from our data. To thisaxis. A faint parabola is also visible in the experimental data end, we ﬁrst extract the experimental band dispersion from
´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505 501Fig. 5. Binding energy proﬁles for Be extracted from the measured and calculated energy–momentum resolved density. Calculations arespherically averaged LDA with the Monte Carlo simulation. A smooth background has been subtracted from all data sets. Calculated bindingenergy and intensity of the valence band has been normalised to the experiment at 0.0 a.u. momentum.data similar to that in Fig. 5. The data were binned into a ﬁner Table 2(0.05 a.u.) momentum interval and the least squares ﬁtting Experimental and calculated Be conduction band widths (eV).method described earlier used to extract the peak positions. Theoretical predictions based upon experimental lattice constantsThe result is shown in Fig. 3. Beyond about 1 a.u. in Method/basis Bandwidth (eV)momentum, the binding energy proﬁles of Fig. 5 arerelatively small and very broad and it is difﬁcult to extract HF/5-111G 19.52meaningful peak positions. The experimental points plotted LDA/5-111G 11.55in Fig. 3 beyond 1 a.u. correspond to the energy of the middle PBE/5-111G 11.7of the peak. For this reason the experimental (spherically PBE0/5-111G 13.65averaged) bandwidth in Fig. 3 appears much smaller than the HF/vtz1 19.24calculated bandwidths. To obtain the full bandwidth requires LDA/vtz1 10.97knowing the work function of the spectrometer, since our PBE/vtz1 11.13binding energies are measured relative to this point. This can PBE0/vtz1 13.24be accomplished by comparing the experimental and Experiment 11.1 ^ 0.3theoretical binding energy proﬁles integrated over
502 ´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505a 0.8 – 2 a.u. range. The two proﬁles have the same shape but the experimental error. Inclusion of exact exchange inare shifted in energy, and by normalising the observed and the PBEO hybrid method gives values which are aroundcalculated energies at the G-point, this shift provides 2 eV wider than experiment.a reasonable estimate of the spectrometer work function. Previous experimental measurements of the bandwidthWith this method, we obtain a value of 3.2 ^ 0.2 eV. Our range from 10.25 ^ 0.5 eV from the X-ray emission data ofmeasured energy at the G-point relative to the vacuum level Lukirskii and Brytov  to 16.88 eV from the k-emissionof the spectrometer is 14.3 ^ 0.2 eV, hence the energy spectrum of Sagawa . Using the same technique,relative to the Fermi level (or bandwidth) is 11.1 ^ 0.30 eV Skinner  reported a value of 13.8 ^ 1 eV. More Comparing the data of Table 2, HF overestimates the recently, in an ARPES measurement, Jensen et al. bandwidth by nearly a factor of 2 irrespective of the size obtain a value of 11.1 ^ 0.1 eV.of the basis set used. The larger vtz1 basis set narrowsthe bandwidths with respect to the 5-111G basis by less 4.2. BeOthan 1 eV for all methods used. Although these resultsare in some ways expected, it is also surprising given the The experimental (points with error bars) and sphericallypopularity of HF for predicting optimum crystal struc- averaged LDA (solid line) binding energy proﬁles aretures, and the noticeable superior performance of the shown in Fig. 6. A smooth background has been subtractedextended vtz1 basis set in this regard. Clearly a basis set from the experimental data with a third order polynomial tooptimised at the HF level to reproduce physical crystal aid comparison with the calculation: the Monte Carlostructures may not necessarily reproduce electronic simulation (dotted line) is dominated by elastic scatteringstructures. With the inclusion of electron correlation, all events and does not reproduce this background.three DFT calculations give signiﬁcantly narrower The spherically averaged LDA calculation reproducesbandwidths that are more in line with the observation, the measured intensity and dispersion of the lower O(2s)although technically, DFT should not be used for this band relatively well, but does not perform so well for thetype of comparison. The LDA and gradient corrected upper O(2p) band, particularly in regard to the intensity(PBE) methods give best agreement, with the PBE/vtz1 distribution. The calculation predicts very little intensity atcombination reproducing the experimental value to within the G-point, whereas we observe considerable intensity.Fig. 6. Binding energy proﬁles for BeO extracted from the EMS measurement and calculated energy–momentum resolved density. Calculationsare spherically averaged LDA, with and without the Monte Carlo simulation of multiple scattering. A smooth background has been subtractedfrom the experimental data. Calculated binding energy and intensity of the 2s band has been normalised to the experiment at zero momentum.The calculated O(2p) peak at 0 and 0.1 a.u. momentum with no Monte Carlo simulation has been scaled upwards by a factor of 10.
´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505 503Small angle elastic scattering which smears intensity along consistent with these values. We obtain thethe momentum axis will contribute to intensity at the experimental value from a least squares ﬁt to the EMSG-point, however, our simulation of this process using the binding energy proﬁles similar to those shown in Fig. 6, butMonte Carlo procedure does not produce sufﬁcient intensity with a 0.05 a.u momentum integration. Although it is ato explain the observation. In addition, the shape of the spherical average, this value is comparable to previousmeasured and predicted binding energy proﬁles are quite values due to degeneracy at the G-point.different. At higher momentum values, the Monte Carlo The DOS bandgaps can be compared directly with thesimulation reproduces the observed proﬁles well, albeit with ones derived from our calculated dispersion curves similarmuch smaller intensity. The intensity ratio between the to Fig. 3. Values are given by the ﬁrst number in eachupper and lower valence bands across the entire momentum column of Table 3. HF overestimates the bandgap comparedrange also appears quite different in the experiment with the X-ray and photoemission measurements. The DFTcompared with the calculation, and again cannot be methods give much better agreement, with LDA and PBEremedied by including multiple scattering effects in the underestimating and PBEO giving perhaps the closestcalculation. agreement. The plane-wave pseudopotential calculation of Although it is clear that multiple scattering, particularly Chang et al.  predicts a value of 19 eV.elastic scattering, does re-distribute intensity in the bands, it Values from our spherically averaged calculationis not sufﬁcient to explain the large discrepancy between obtained by ﬁtting the predicted binding energy proﬁlesexperiment and theory at small momentum values. It also are also given in Table 3 in parentheses. Comparing theseseems unlikely that the ﬁnite momentum resolution of the values with our experiment, PBE gives the closestspectrometer is responsible for this difference, as this has, to prediction. Spherical averaging reduces the predictedsome degree, been built into the binding energy proﬁles bandgap by only about 1%. It is therefore reasonable tothrough the momentum integration. The anomalous inten- compare our experimental value directly with previous DOSsity may indicate a contribution of cation states to the values. This slight change in bandgap arises because therepredominantly oxygen valence band or that the ionisation are two separate O(2p) bands at the G-point: most of theprocess itself has some momentum dependence to it. The intensity lies in the upper of these bands but the smallcovalent character of BeO is well known and experimental amount present in the other shifts the spherical average to aand theoretical studies point to the importance of including lower value.metal ion p-states at the top of the valence band. It seems The above X-ray emission and photoemission measure-unlikely that the scattering cross-section is the origin of ments give O(2p) bandwidths that range from 8.8  tothese differences, as this is believed to be well understood 10.3 eV ; our experimental value, on the other hand isfrom experiments on atomic and molecular targets. Never- 4.7 ^ 0.2 eV. Care has to be taken comparing these valuestheless, we ﬁnd that LCAO calculations predict other because of the effects of spherical averaging, whichaspects of the EMS electronic structures, such as bandgaps becomes clear when comparing the calculated bandwidthsand bandwidths reasonably well. in Table 3. Spherical averaging reduces the O(2p) Oxygen intervalence bandgaps at the G-point have bandwidth by about 17% due to the bands being non-been obtained in previous X-ray and photoemission degenerate along different directions at the band minimum.measurements of the density of states (DOS). Fomichev However, the experimental bandwidth is still narrower than reports a value of 21.3 eV, and Lukirskii and the spherically averaged calculations, with HF overestimat-Brytov , 19.5 eV. Photoemission measurements of ing by about 40%, and PBE giving the smallest overestimateHamrin et al.  give a similar value of 20 eV. Our at 10%.experimental result of 19 ^ 0.2 eV (listed in Table 3) is Comparing our calculated bandwidths, derived directly from dispersion curves for BeO, with previously published X-ray and photoemission measurements, the HF calculationTable 3 underestimates the O(2p) bandwidth from the experiments.Measured and predicted bandwidths for BeO in eV. The ﬁrst Including electron correlation effects with the DFT methodsnumber in each column is calculated from the dispersion curve, produces even narrower bandwidths. This is in contrast tonumbers in brackets are from the spherically averaged calculations metallic Be, where bandwidths are generally overestimatedMethod G1 ! G1,6 DE by HF and in reasonable agreement with DFT calculations. Two previous DFT calculations [18,34] both give a O(2p) O(2s) bandwidth of 6 eV. A later X-ray emission measurement of the DOS  gives a value of 6.8 eV for the bandwidth,HF (sph av) 25.22 (24.93) 8.05 (6.77) 2.84 (2.56) which is more consistent with our calculation andLDA (sph av) 18.47 (18.29) 6.43 (5.39) 2.39 (2.19) considerably narrower than the previous X-ray and photo-PBE (sph av) 18.73 (18.50) 6.36 (5.30) 2.32 (2.09) emission measurements. The difﬁculty of comparingPBE0 (sph av) 20.44 (20.22) 6.88 (5.75) 2.47 (2.27)Expt 19 ^ 0.2 4.7 ^ 0.2 1.7 ^ 0.2 photoemission DOS measurements with calculations has been pointed out previously , where it was suggested
504 ´ B. Soule de Bas et al. / Journal of Physics and Chemistry of Solids 64 (2003) 495–505that bandwidths extracted from the photoemission DOS may measurements, we ﬁnd that all three of our DFT calculationsbe broadened by instrumental and thermal effects. slightly overestimate the widths of the oxygen valence From our calculations, it is obvious that spherically bands in BeO. For both the 2p and 2s bands, PBE gives theaveraging reduces the O(2s) bandwidth, in this case by narrowest width followed by LDA then PBEO. Previousabout 10%. Once again our measured value falls below calculations agree with our predictions and the more recentall the calculations with PBE giving a value just outside the X-ray emission measurement gives a narrower 2p band-experimental error. Previous experimental measurements width that better agrees with the EMS result.do not assign a width to the O(2s) band. The PBE calculation gives excellent agreement for the bandwidth in Be provided the extended basis set is employed. With the 5-111G basis, all the DFT methods overestimate the bandwidth, LDA being narrowest and5. Conclusions PBEO considerably broader. For the extended basis, LDA underestimates and PBEO overestimates the bandwidth. In this paper we have presented EMS of Be and BeO. Overall, the PBE functional within a DFT frameworkEMS is a technique that provides a direct and complete reproduces the measured band energies most accurately,mapping of band structures. The Be and BeO results are although it must be remembered that strictly speaking DFTrelatively free of the effects of multiple scattering and so only provides the total ground state energy and density andcan provide a benchmark set of data for the technique not the quasi-particle energies.and a quantitative comparison with theoretical models. The experimental band intensities, of both Be and BeOWe have also presented LCAO electronic structure are signiﬁcantly different to model predictions of any level.calculations within the HF and DFT formalisms. For the metal, the discrepancies arise from many-bodySpherical averaging over all crystal directions (to account interactions in the form of electron – hole lifetimes.for the polycrystalline structure of the target) and However, the differences in the oxide are not readilymultiple scattering effects have been included in our explained. Our Monte Carlo simulations indicate thatmodelling efforts in order to address the differences multiple scattering effects are not the primary cause ofbetween theory and observation. anomalous intensity. One explanation could be the inability The HF method is very popular and has proved quite of the model calculations to adequately describe covalentsuccessful for modelling crystal structures. However, we effects in the bonding.ﬁnd strong disagreement between EMS measurements The observed discrepancies between theory and exper-and HF predictions of electronic structure. For Be metal iment, particularly the difference in O(2p) band intensityin particular, HF overestimates the bandwidth by nearly a and bandwidth for the oxide, warrant further investigation.factor of 2. Clearly, inclusion of electron correlation is The thickness of the target used in these measurements maycritical in modelling the electronic structures of both the be responsible for these differences. Our future efforts willmetallic and ionic species. The use of an extended basis concentrate on understanding the character of the targetsset for Be, for its high computational overhead, does not used in the experiment and incorporating this into thesigniﬁcantly improve electronic structures at the HF modelling, in particular, calculating band structures forlevel, although its predicted lattice constants are in slabs rather than bulk since the targets are relatively thin.excellent agreement with experiment. All three levels of DFT calculation reproduce theEMS results more closely. In the case of the intervalence Acknowledgementsbandgap in BeO, both LDA and gradient corrected DFTunderestimate the experimental value, although gradient This work was supported by the Australian Researchcorrections (PBE) give larger values. Inclusion of exact ´ Council and Flinders University. B. Soule de Bas thanks theexchange (PBEO) increases the bandgap further to the Flinders Institute for Research in Science and Technologypoint of overestimating the experimental value. The for providing support as a Visiting Scholar.intervalence bandgap is relatively unaffected by sphericalaveraging since most of the intensity lies in degeneratebands along different directions at the G-point. Therefore,our EMS bandgaps can be compared directly with Referencesphotoemission or X-ray measurements and other  M.A. Coplan, J.H. Moore, J.P. Doering, Rev. Mod. Phys. 66calculations. Our results are consistent with previously (1994) 985.published values.  I.E. McCarthy, E. Weigold, Rep. Prog. Phys. 54 (1991) 789. Even accounting for the effects of spherical averaging in  M.J. Ford, H.E. Dorsett, M.J. Brunger, Trends in Appl.our data, earlier X-ray and photoemission O(2p) bandwidths Spectrosc. 3 (2001) 145.are considerably wider than either our calculated or  J.R. Dennison, A.L. Ritter, J. 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