The electronic structure of Be and BeO: benchmark EMS measurements and LCAO calculations

Abstract

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 electron multiple scattering within the sample, resulting in very clean "benchmark" measurements for the EMS technique. Experimental data are compared to tight-binding (LCAO) electronic structure calculations using Hartree-Fock (HF), and local density (LDA-VWN), gradient corrected (PBE) and hybrid (PBE0) density functional theory. Overall, DFT calculations reproduce the EMS data for metallic Be reasonably well. PBE predictions for the valence bandwidth of Be are in excellent agreement with EMS data, provided the calculations employ a large basis set augmented with diffuse functions. For BeO, PBE calculations using a moderately-sized basis set are in reasonable agreement with experiment, slightly underestimating the valence bandgap and overestimating the O(2s) and O(2p) bandwidths. The calculations also underestimate the EMS intensity of the O(2p) band around the Γ-point. Simulation of the effects of multiple scattering in the calculated oxide bandstructures do not explain these systematic differences.

Electronic band structure of calcium oxide

“…This discrepancy seems to be due to the failure of the LDA approximation to correctly describe electron correlation effects in insulating solids. More sophisticated schemes, like GCA and hybrid approximations, give a better agreement with the experiment [49]. The LMTO calculation predicts no intensity at the Γ point, typical for p-like orbitals, but the experiment shows an intense broad structure of a width of approximately 2.5 eV with a height even larger than that of the s-band peak at 0 a.u..…”

“…However we do not recommend this in the LDA calculations. Multiple scattering effects were implemented into LCAO calculations at the GGA and hybrid level and compared with our BeO results [49] but multiple scattering does not completely explain all the discrepancies.…”

Trends in the band structures of the group-I and -II oxides

“…This is not surprising given that the basis sets are optimised at the To compare our calculations directly with the EMS data, we must generate a spherically averaged energy-momentum map from the calculation. We achieve this by calculating along 25 evenly spaced crystal directions, which we find is sufficient to give a very good representation of the true spherical average over all crystal directions 24 . The calculated band dispersion relation and electron momentum density (EMD) for each of the 25 directions are combined to give the energy and momentum resolved probability density, and then summed The calculated, spherically averaged, energy-momentum map can now be analysed in exactly the same manner as the experimental data for a direct comparison.…”

“…Studies using Monte Carlo simulations to generate a multiple scattering background for calculated spectra 24 indicate that multiple scattering has only a small effect on band energies derived from EMS data. The least squares fit to the background subtracted data is also shown.…”

“…While these effects can, to some extent, be compensated in lattice parameter calculations (and other elastic properties) by using extended basis sets, it is unlikely that a Hamiltonian without correlation can provide reliable electronic structure predictions 24,25 . It is also clear that there is currently insufficient experimental evidence with which to test these aspects of any calculations.…”