How To Get DFT To Work for CeOHow To Get DFT To Work for CeO22

Christopher CastletonChristopher Castleton11, Jolla Kullgren, Jolla Kullgren22, Carsten Müller, Carsten Müller33, , David Muñoz RamoDavid Muñoz Ramo44 , Amy Green , Amy Green11 & Kersti Hermansson & Kersti Hermansson22

1. Nottingham Trent University, UK1. Nottingham Trent University, UK2. Uppsala University, Sweden2. Uppsala University, Sweden

3. Free University Berlin, Germany3. Free University Berlin, Germany4. University College London, UK4. 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 (CeOCerium Dioxide (CeO22, Ceria), Ceria)

Flourite Structure

Uses: Catalysis (esp car exhausts) Fuel cellsGas sensorsReplacement 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 CeriaBands 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 CeriaBands 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 CeriaBands 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 CeOWhy is CeO22 Awkward? 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.”

Supercells: Choice often controls the results….

Functionals: LDA & GGA fail!

1: Kresse et al. PRB 72, 237101 (2005) 2: Kullgren et al. JCP 132, 054110 (2010)

Localization: Many possible….

Neutral VNeutral VOO: LDA vs LDA+U: 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)

CeOCeO22: LDA+U vs GGA+U: 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

• Mix Hartree Fock (exact exchange, NO correlation) with LDA/GGA (partial exchange, partial correlation).

• B3LYP has been popular for molecules, so … Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1

Hybrid FunctionalsHybrid Functionals

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 properties11: Kullgren et al. JCP 132,

054110 (2010)

Polarons in Ceria (II)Polarons in Ceria (II)

Can we study the dynamics of “free” polarons?

Expt: ~0.5 eV

Using LDA+U => Maybe ….1

1: Castleton, Green et al. in preparation.

Ceria (110) surface vacanciesCeria (110) surface vacancies

Kullgren et al. Submitted to JCP

• Oxygens on (110) are paired.

• Remove one, & the other moves, but previous authors didn’t agree how.

• Oxygens on (110) are paired.

• Remove one, & the other moves, but previous work didn’t agree how.

Localization PatternsLocalization Patterns

Kullgren et al. Submitted to JCP

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 => Bridge

•1+3, 1+4, 1+8, 3+5, => Distorted

•1+2 metastable alternative => In plane

1+4

1

4

New Problem:•Energies of some patterns change by up to ~1 eV between different supercells.

Problems with Supercells.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.Assessing Supercell Errors.

• Can assess errors by scaling with 1/L (supercell size).

VASP, PAW, LDA, 200 eV planewave cut

Castleton et al. Modeling Simul. Mater. Sci. Eng. 17 084003 (2009)

• In bulk: Errors ~ 1/L (length) & ~ 1/L3 (volume)• Effect on energetics can be VERY significant.

• Can’t manage p(4x4) supercell, so must

• Check effect of defect images in x and y directions separately:

Ceria (110) surface supercellsCeria (110) surface supercells

Errors in p(2x2): X => 0.2 – 1.0 eV y => 0.0 – 0.1 eV

When combined, the optimal localization changes:

1

4

3

4

Functionals:• LDA+U is best.

• For hybrids B3LYP is better for electronic propertiesPBE0 is better for structural properties

SummarySummary

Surface Vacancies & Polarons:• Electron localization especially complex.

• Supercells of a few 10s or 100s of atoms often cause errors of several eV.

CeO2

• Interesting, but hard to treat, due to Ce 4f electron localization both times!

Core Electrons:• No US-PP, need small core PAW

• 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.

Core ElectronsCore Electrons

• PAW: 46 electrons in the core has a ghost state: need 28 electron ECPs.

Band gaps U valuesBand gaps U values

InP Defects ScalingInP Defects Scaling

InP Defects Scaling InP Defects Scaling (Neutral)(Neutral)

CeOCeO22: LDA vs LDA+U: 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!

• Mix Hartree Fock (exact exchange, NO correlation) with LDA/GGA (partial exchange, partial correlation).

• B3LYP has been popular for molecules, so … Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1

Hybrid FunctionalsHybrid Functionals

CeO2

Ce2O3

1: Kullgren et al. JCP 132, 054110 (2010)

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

• Mix Hartree Fock (exact exchange, NO correlation) with LDA/GGA (partial exchange, partial correlation).

Hybrid FunctionalsHybrid Functionals

• B3LYP has been popular for molecules, so … Question: Does B3LYP work OK for ceria, & hence for molecules on ceria?1

• LDA+U does best.1 • B3LYP is better hybrid for electronic properties1

• PBE0 is better hybrid for structural properties11: Kullgren et al. JCP 132,

054110 (2010)