Tuning LDA+U for electron localization and structure at oxygen vacancies in ceria

Abstract

We examine the real space structure and the electronic structure ͑particularly Ce4f electron localization͒ of oxygen vacancies in CeO 2 ͑ceria͒ as a function of U in density functional theory studies with the rotationally invariant forms of the LDA+ U and GGA+ U functionals. The four nearest neighbor Ce ions always relax outwards, with those not carrying localized Ce4f charge moving furthest. Several quantification schemes show that the charge starts to become localized at U Ϸ 3 eV and that the degree of localization reaches a maximum at ϳ6 eV for LDA+ U or at ϳ5.5 eV for GGA+ U. For higher U it decreases rapidly as charge is transferred onto second neighbor O ions and beyond. The localization is never into atomic corelike states; at maximum localization about 80-90% of the Ce4f charge is located on the two nearest neighboring Ce ions. However, if we look at the total atomic charge we find that the two ions only make a net gain of ͑0.2-0.4͒e each, so localization is actually very incomplete, with localization of Ce4f electrons coming at the expense of moving other electrons off the Ce ions. We have also revisited some properties of defect-free ceria and find that with LDA+ U the crystal structure is actually best described with U =3-4 eV, while the experimental band structure is obtained with U =7-8 eV. ͑For GGA+ U the lattice parameters worsen for U Ͼ 0 eV, but the band structure is similar to LDA + U.͒ The best overall choice is U Ϸ 6 eV with LDA+ U and Ϸ5.5 eV for GGA+ U, since the localization is most important, but a consistent choice for both CeO 2 and Ce 2 O 3 , with and without vacancies, is hard to find.

Disentangling the role of small polarons and oxygen vacancies in CeO2

“…Some researchers reported that the two excess electrons prefer to localize on two Ce atoms that are the nearest neighbor (NN) to the vacancy [15,16,29,30,[34][35][36][37][38][39], while others [40,41,42] suggested that having the Ce 3+ on the next-nearest neighbor (NNN) sites is energetically more favorable over the NN configuration.…”

“…The results of DFT+U studies are largely dependent on the U value, bringing some uncertainty to the results and raising questions on what value of U should be used to describe the properties of perfect and reduced CeO 2 . Furthermore, the band gap and the oxygen-vacancy formation energy derived from DFT+U calculations have noticeable differences from experimental data [43,44], with the reported oxygen-vacancy formation energies varying over a wide range, from 2.3 to 4.7 eV [10,13,16,[29][30][31][32][33][34][35][36][37][38][39].…”

“…For the oxygen vacancy, the excess electrons are localized on individual Ce atoms, turning two Ce 4+ into two Ce 3+ species, which, by Coulomb attraction, sit near to the vacancy [29][30][31][32][33][34][35][36][37][38][39][40][41][42]. The results of DFT+U studies are largely dependent on the U value, bringing some uncertainty to the results and raising questions on what value of U should be used to describe the properties of perfect and reduced CeO 2 .…”

“…The DFT+U method has also been employed to study the electronic properties of CeO 2 and the impact of oxygen vacancies [29,30,[33][34][35][36][37][38][39][40][41][42]. In this approach, an extra on-site Coulomb interaction is added to the Ce 4f states, leading to a split between occupied and unoccupied states.…”

“…There have appeared numerous studies of bulk CeO 2 based on DFT calculations, many focusing on the oxygen vacancy [10][11][12][13][14][15][16][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. DFT calculations using the local density approximation (LDA) or the generalized gradient approximation (GGA) for the exchange-correlation term underestimate the separation between the top of the valence band (O 2p band) and the empty Ce 4f bands in CeO 2 by ~35 % [16,[31][32][33].…”