1. How To Get DFT To Work for CeO 2 Christopher Castleton 1, Jolla Kullgren 2, Carsten Müller 3, David Muñoz Ramo 4, Amy Green 1 & Kersti Hermansson 2 1. Nottingham Trent University, UK 2. Uppsala University, Sweden 3. Free University Berlin, Germany 4. University College London, UK  Why is CeO 2 interesting?  Why is CeO 2 awkward?  What can we do about it….  …. and a little of what we have done about it!

2. Cerium Dioxide (CeO 2, Ceria) Flourite Structure  Uses: Catalysis (esp car exhausts) Fuel cells Gas sensors Replacement for SiO 2 in CMOS  V O Most common defect Create & remove easily => “oxygen storage capacity”. Anticipate double donor => V O +2 …… but actually form as neutral V O +0 …..  Ce 2 O 3 Can continuously remove oxygen to reach Ce 2 O 3.

3. 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.

4. 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.  V O +0 “Donated” electrons stay very local. VOVO

5. 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.  V O +0 “Donated” electrons stay very local.  Ce 2 O 3 One localized Ce:4f electron per Ce.

6. Why is CeO 2 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….

7. Neutral V O : LDA vs LDA+U Pure LDA:Ce4f electrons not localized. LDA+U ≈ 6eV, Ce4f localized, level in gap 1. U choice is a compromise: Ce 2 O 3 & some CeO 2 gaps want smaller U, other gaps larger… 1: Castleton et al. JCP 127, 244704 (2007)

8. CeO 2 : 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

9. 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 Functionals CeO 2 a0a0 B PBE05.411220 B3LYP5.475199 LDA+U5.405213 Expt.5.391204-230 0 K values LDA+Udoes best. 1 B3LYPis better hybrid for electronic properties 1 PBE0is better hybrid for structural properties 1 1: Kullgren et al. JCP 132, 054110 (2010)

10. 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.

11. Ceria (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.

12. Oxygens on (110) are paired. Remove one, & the other moves, but previous work didn’t agree how. Localization 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.

13. 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?

14. 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/L 3 (volume) Effect on energetics can be VERY significant.

15. Can’t manage p(4x4) supercell, so must Check effect of defect images in x and y directions separately: Ceria (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

16. Functionals: LDA+U is best. For hybridsB3LYP is better for electronic properties PBE0 is better for structural properties Summary Surface Vacancies & Polarons: Electron localization especially complex. Supercells of a few 10s or 100s of atoms often cause errors of several eV. CeO 2 Interesting, but hard to treat, due to Ce 4f electron localization both times! Core Electrons: No US-PP, need small core PAW

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

18. Band gaps U values

19. InP Defects Scaling

20. InP Defects Scaling (Neutral)

21. CeO 2 : LDA vs LDA+U Pure LDA:Ce4f electrons not localized. LDA+U ≈ 6eV, Ce4f localized, level in gap. Choice not unique: Ce 2 O 3 & some CeO 2 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!

22. 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 Functionals CeO 2 Ce 2 O 3 1: Kullgren et al. JCP 132, 054110 (2010)

23. CeO 2 Ce 2 O 3 CeO 2 a0a0 Ba0a0 c0c0 u Ce PBE05.4112203.8716.0740.2450.646 B3LYP5.4751993.8976.1940.2470.646 LDA+U5.4052133.8566.0430.2460.646 Expt.5.391204-2303.8916.0590.2450.647 0 K valuesR.T. values Mix Hartree Fock (exact exchange, NO correlation) with LDA/GGA (partial exchange,partial correlation). Hybrid Functionals B3LYP has been popular for molecules, so … Question: Does B3LYP work OK for ceria, & hence for molecules on ceria? 1 LDA+Udoes best. 1 B3LYPis better hybrid for electronic properties 1 PBE0is better hybrid for structural properties 1 1: Kullgren et al. JCP 132, 054110 (2010)