356
W. Piskorz and F. Zasada
show the failure of the GGA-PBE or PW91 [103] functional which erroneously
yield a metallic ground state of defected ceria, so do the LDA functionals [145].
The hybrid functionals tend to overestimate the CeO 2 band gap by 45% (PBE0) and
15% (HSE), comparing to experiments [27]. Very reasonable is the pragmatic, albeit
non-universal, DFT+U approach as show by Nolan et al. [105, 146] and Fabris et al.
[145], where the calculations with standard atomic orbitals and the Wannier localised
functions were carried out for reduced (defect concentration of 1 /4) ceria (100), (110),
and (111) surfaces yielding the well agreed surface reconstruction. The obtained
geometries did not significantly depend on the particular variant of the DFT+U
method, contrary to the defect formation energies, and it can be inferred that the Ce
ions closest to the vacancy are repelled from it, while the O anions move towards
the defect site [145]. The value of U, however, needs to be adjusted accordingly to
the required property. For example, to obtain the energy of CO adsorption on CeO 2
the value of U was found to be 2 eV [147], while for the proper reproduction of the
electronic structure of CeO 2 the value of U = 4.5–6.0 eV is required [27, 148, 149].
The use of hybrid functional, e.g. HSE, significantly improves the results [81]
for the reduction of the self-interaction error, which leads to the exaggeration of
electron delocalisation. The electrons occupy two localised Ce 4f states adjacent to
the vacancy and, in the energy terms, ca. 1 eV below the conduction band within
the band gap. According to, e.g. Paier et al. [81], the hybrid functional can be used
for validation of the results obtained by PBE+U and gives results generally closer to
the experiment than PBE+U. Adding the dispersion effects, DFT+D, improves the
results shifting them towards experimental values by 0.12–0.15 eV. Nonetheless, the
deviation from the experiment for the O vacancy formation was still quite high, ca.
1.7 eV for HSE+D or 2.2 eV for PBE+U+D. For the low concentrations of defects,
i.e. for the calculations on the model of supercell 4 × 4 × 4, the HSE functional
underestimates the defect formation by ca. 0.5 eV with respect to the best experimental results available (4.2 ± 0.3 eV [150]), while PBE+U(4.5) underestimates
them by ca. 1.4 eV.
O Diffusion
Another important parameter in the description of redox properties of oxides is the
oxygen diffusion. For CeO 2 , the DFT+U calculations [151] gave the energy barrier of
51 kJ/mol, very close to the experimental value of 50 ± 16 kJ/mol for defect concentration of 4%. The electronic conductivity, also contributing to the ceria conductivity,
has been modelled by small polarons. Panhans et al. [150] in their experiments at
fixed p(O 2 ) found the conductivity to be n-type with resultant activation energy of
1 /2ΔH d ( 1 /2O 2 ) + ΔH
polaron = 2.57 ± 0.33 eV, where ΔH d ( 1 /2O 2 ) is the defect formation energy, and ΔH
polaron is the energy barrier for polaron hopping. The ΔH
polaron
value was found to be 0.48 eV, very close to the experimental value of 0.40 eV [152].
W. Piskorz and F. Zasada
show the failure of the GGA-PBE or PW91 [103] functional which erroneously
yield a metallic ground state of defected ceria, so do the LDA functionals [145].
The hybrid functionals tend to overestimate the CeO 2 band gap by 45% (PBE0) and
15% (HSE), comparing to experiments [27]. Very reasonable is the pragmatic, albeit
non-universal, DFT+U approach as show by Nolan et al. [105, 146] and Fabris et al.
[145], where the calculations with standard atomic orbitals and the Wannier localised
functions were carried out for reduced (defect concentration of 1 /4) ceria (100), (110),
and (111) surfaces yielding the well agreed surface reconstruction. The obtained
geometries did not significantly depend on the particular variant of the DFT+U
method, contrary to the defect formation energies, and it can be inferred that the Ce
ions closest to the vacancy are repelled from it, while the O anions move towards
the defect site [145]. The value of U, however, needs to be adjusted accordingly to
the required property. For example, to obtain the energy of CO adsorption on CeO 2
the value of U was found to be 2 eV [147], while for the proper reproduction of the
electronic structure of CeO 2 the value of U = 4.5–6.0 eV is required [27, 148, 149].
The use of hybrid functional, e.g. HSE, significantly improves the results [81]
for the reduction of the self-interaction error, which leads to the exaggeration of
electron delocalisation. The electrons occupy two localised Ce 4f states adjacent to
the vacancy and, in the energy terms, ca. 1 eV below the conduction band within
the band gap. According to, e.g. Paier et al. [81], the hybrid functional can be used
for validation of the results obtained by PBE+U and gives results generally closer to
the experiment than PBE+U. Adding the dispersion effects, DFT+D, improves the
results shifting them towards experimental values by 0.12–0.15 eV. Nonetheless, the
deviation from the experiment for the O vacancy formation was still quite high, ca.
1.7 eV for HSE+D or 2.2 eV for PBE+U+D. For the low concentrations of defects,
i.e. for the calculations on the model of supercell 4 × 4 × 4, the HSE functional
underestimates the defect formation by ca. 0.5 eV with respect to the best experimental results available (4.2 ± 0.3 eV [150]), while PBE+U(4.5) underestimates
them by ca. 1.4 eV.
O Diffusion
Another important parameter in the description of redox properties of oxides is the
oxygen diffusion. For CeO 2 , the DFT+U calculations [151] gave the energy barrier of
51 kJ/mol, very close to the experimental value of 50 ± 16 kJ/mol for defect concentration of 4%. The electronic conductivity, also contributing to the ceria conductivity,
has been modelled by small polarons. Panhans et al. [150] in their experiments at
fixed p(O 2 ) found the conductivity to be n-type with resultant activation energy of
1 /2ΔH d ( 1 /2O 2 ) + ΔH
polaron = 2.57 ± 0.33 eV, where ΔH d ( 1 /2O 2 ) is the defect formation energy, and ΔH
polaron is the energy barrier for polaron hopping. The ΔH
polaron
value was found to be 0.48 eV, very close to the experimental value of 0.40 eV [152].