How To Get DFT To Work for CeO2
Christopher Castleton1, Jolla Kullgren2, Carsten Müller3,
David Muñoz Ramo4 , Amy Green1 & Kersti Hermansson2
1. Nottingham Trent University, UK
2. Uppsala University, Sweden
3. Free University Berlin, Germany
4. University College London, UK
 Why is CeO2 interesting?
 Why is CeO2 awkward?
 What can we do about it….
 …. and a little of what we have done about it!
Cerium Dioxide (CeO2, Ceria)
Flourite Structure
 Uses:
Catalysis (esp car exhausts)
Fuel cells
Gas sensors
Replacement for SiO2 in CMOS
 VO
Most common defect
Create & remove easily
=> “oxygen storage capacity”.
Anticipate double donor => VO+2
…… but actually form as neutral VO+0…..
 Ce2O3
Can continuously remove oxygen to reach Ce2O3.
Bands and Polarons in Ceria
 Bands:
Empty Ce:4f band lies between valence & conduction.
 Localization: Electrons entering this localize on individual Ce ions.
=> Local lattice distortion => a self trapped “polaron”.
 Conductivity: Thermally activated polaron “hopping” …….. not band-like.
Bands and Polarons in Ceria
 Bands:
Empty Ce:4f band lies between valence & conduction.
 Localization: Electrons entering this localize on individual Ce ions.
=> Local lattice distortion => a self trapped “polaron”.
 Conductivity: Thermally activated polaron “hopping” …….. not band-like.
 VO+0
“Donated” electrons stay very local.
VO
Bands and Polarons in Ceria
 Bands:
Empty Ce:4f band lies between valence & conduction.
 Localization: Electrons entering this localize on individual Ce ions.
=> Local lattice distortion => a self trapped “polaron”.
 Conductivity: Thermally activated polaron “hopping” …….. not band-like.
 VO+0
“Donated” electrons stay very local.
 Ce2O3
One localized Ce:4f electron per Ce.
Why is CeO2 Awkward?
 Core e- :
US-PP fails!1 Need PAW, with relatively hard Ce potential
or a small core ECP with an atomic basis.2
Otherwise you get “ghost states.”
 Functionals: LDA & GGA fail!
 Supercells:
Choice often controls the results….
 Localization: Many possible….
1: Kresse et al. PRB 72, 237101 (2005) 2: Kullgren et al. JCP 132, 054110 (2010)
Neutral VO: LDA vs LDA+U
• Pure LDA:
Ce4f electrons not localized.
• LDA+U ≈ 6eV, Ce4f localized, level in gap1.
• U choice is a compromise:
Ce2O3 & some CeO2 gaps want smaller U,
other gaps larger…
1: Castleton et al. JCP 127, 244704 (2007)
CeO2: LDA+U vs GGA+U
• For ceria, GGA worsens LDA’s balance of correlation & exchange errors so
LDA beats GGA & LDA+U beats GGA+U
Hybrid Functionals
• Mix Hartree Fock
with LDA/GGA
(exact exchange, NO correlation)
(partial exchange, partial correlation).
• B3LYP has been popular for molecules, so …
Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1
CeO2
CeO2
a0
B
PBE0
5.411
220
B3LYP
5.475
199
LDA+U
5.405
213
Expt.
5.391
204-230
0 K values
• LDA+U does best.1
• B3LYP is better hybrid for electronic properties1
• PBE0 is better hybrid for structural properties1
1: Kullgren et al. JCP 132,
054110 (2010)
Polarons in Ceria (II)
 Can we study the dynamics of “free” polarons?
Using LDA+U => Maybe ….1
Expt:
~0.5 eV
1: Castleton, Green et al. in preparation.
Ceria (110) surface vacancies
• Oxygens on (110) are paired.
• Remove one, & the other moves, but previous authors didn’t agree how.
Kullgren et al. Submitted to JCP
Localization Patterns
• Oxygens on (110) are paired.
• Remove one, & the other moves, but previous work didn’t agree how.
The problem was, where do the Ce 4f electrons localize.
=> many possibilities, all within about ~0.5 eV!
•1+2, 3+4, 3+6, 4+9, 7+8
•1+3, 1+4, 1+8,
1+4 3+5,
•1+2 metastable alternative
=> Bridge
=> Distorted
=> In plane
4
New Problem:
•Energies of some patterns change by up to ~1
eV between different supercells.
1
Kullgren et al. Submitted to JCP
Problems with Supercells.
• NOT doing defect calculations.
=> an infinite, ordered array of interacting “defects”.
Question:
How big are the errors compared to infinite supercell / lone vacancy?
Assessing Supercell Errors.
VASP, PAW,
LDA, 200 eV
planewave cut
• Can assess errors by scaling with 1/L (supercell size).
• In bulk: Errors ~ 1/L (length) & ~ 1/L3 (volume)
• Effect on energetics can be VERY significant.
Castleton et al. Modeling Simul. Mater. Sci. Eng. 17 084003 (2009)
Ceria (110) surface supercells
• Can’t manage p(4x4) supercell, so must
• Check effect of defect images in x and y directions separately:
Errors in p(2x2): X => 0.2 – 1.0 eV
y => 0.0 – 0.1 eV
When combined, the optimal localization changes:
4
1
4
3
Summary
CeO2
• Interesting, but hard to treat, due to Ce 4f electron localization both times!
Core Electrons:
• No US-PP, need small core PAW
Functionals:
• LDA+U is best.
• For hybrids
B3LYP is better for electronic properties
PBE0 is better for structural properties
Surface Vacancies & Polarons:
• Electron localization especially complex.
• Supercells of a few 10s or 100s of atoms often cause errors of several eV.
Core Electrons
• Treat core electrons only as an average field (PAW or ECP)
• ECP: 46 electrons in the core has a ghost state: need 28 electron ECPs.
• PAW: 46 electrons in the core has a ghost state: need 28 electron ECPs.
Band gaps U values
InP Defects Scaling
InP Defects Scaling
(Neutral)
CeO2: LDA vs LDA+U
• Pure LDA: Ce4f electrons not localized.
• LDA+U ≈ 6eV, Ce4f localized, level in gap.
• Choice not unique: Ce2O3 & some CeO2
gaps want smaller, other gaps larger…
• Sledgehammer method: Messes with other
electrons & worsens O2p => Ce5d gap.
• Pure LDA is best if have no Ce4f electrons!
CeO2
Hybrid Functionals
• Mix Hartree Fock
with LDA/GGA
(exact exchange, NO correlation)
(partial exchange, partial correlation).
• B3LYP has been popular for molecules, so …
Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1
Ce2O3
1: Kullgren et al. JCP 132, 054110 (2010)
Hybrid Functionals
• Mix Hartree Fock
with LDA/GGA
(exact exchange, NO correlation)
(partial exchange, partial correlation).
• B3LYP has been popular for molecules, so …
Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1
CeO2
Ce2O3
CeO2
a0
B
a0
c0
uCe
PBE0
5.411
220
3.871
6.074
0.245
0.646
B3LYP
5.475
199
3.897
6.194
0.247
0.646
LDA+U
5.405
213
3.856
6.043
0.246
0.646
Expt.
5.391
204-230
3.891
6.059
0.245
0.647
0 K values
R.T. values
• LDA+U does best.1
• B3LYP is better hybrid for electronic properties1
• PBE0 is better hybrid for structural properties1
1: Kullgren et al. JCP 132,
054110 (2010)