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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 1 Challenges for ab initio defect modeling Peter Deák, Bálint Aradi, and Thomas Frauenheim Bremen Center for Computational Materials Science, University of Bremen POB 330440, 28334 Bremen, Germany Adam Gali Dept. Atomic Physics, Budapest University of Technology & Economics H-1521 Budapest, Hungary

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 2 How good is defect theory today? Challenging some illusions! Richard P MessmerGeorge D Watkins James W Corbett control

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 3 Shishkin&Kresse, PRB 75, 235102 (2007) BAND GAP “State of the art” Supercell: Nr. of Atoms64128216256 Nr. of k points4343 2323 11 Plane Waves (with UPP or PAW ) up to ~240 eV. DFT-GGA: (PBE functional) WHAT COULD POSSIBLY GO WRONG? commercial or public domain „turn-key“ package ?? & GAP STATES! Deák et al.. PRB 75, 153204 (2007) Scissor works only for defects in the high electron density region of the perfect crystal. “the scissor” eVeV eCeC eVeV eCeC

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 4 SiC:V Si U. Gerstmann, P. Deák, et al. Physica B 340-342, 190 (2003). C.-O. Ambladh, U. von Barth, PRB 31, 3231 (1985) Using a correct asymptotic form of the exact exchange correlation potential it is shown that the eigenvalue of the uppermost occupied orbital equals the exact ionization potential of a finite system (atom, molecule, or a solid with a surface). C.-O. Ambladh, U. von Barth, PRB 31, 3231 (1985) Using a correct asymptotic form of the exact exchange correlation potential it is shown that the eigenvalue of the uppermost occupied orbital equals the exact ionization potential of a finite system (atom, molecule, or a solid with a surface). Popular misapprehensions 1.GGA is always successful in describing the ground state of a system. 2.Internal ionization energies (charge transition levels) of defects can be calculated accurately as difference between total energies. 0 Total energies (w.o. gap error) ASSUMPTIONS Problem of charged supercells can be handled by the Makov-Payne correction [ PRB 51, 4014 (1995) ]. Considering vertical transitions (no relaxation of ions) as in optical absorption experiments: Kohn-Sham levels (w.gap error) eDeD eVeV eCeC EgEg The total energy is not affected by the “gap error”! Cancellation??

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 5 Hybrid functionals as etalon 1.M. St ä dele et al., PRB 59, 10031 (1999): “ most of the gap error disappears when using exact exchange in DFT ”. e D -e V LDAG0W0Hybrid Si: H BC (0)+0.61+1.05+1.08 Si: H AB (-)-0.07+0.10+0.05 4H-SiC: H AB (-)+0.50+0.62+0.54 3C-SiC: B Si +2C i (+)-0.12+0.04+0.10 3C-SiC: B Si +2C i (-)+0.18+0.26+0.29 Defect levels 2.A. D. Becke, JCP 107, 8554 (1997): “ mixing HF-exchange to DFT improves calculated molecular properties ” 3.J. Muscat et al, Chem. Phys. Lett. 342, 397 (2001): “ in solids the gap improves as well ”. LATTICE CONSTANT BULK MODULUS COHESIVE ENERGY BAND GAP 4.M. Marsman et al., J. Phys.: Condens. Matter. 20, 064201 (2008): 0.12HF + 0.88PBEExptl. EgEg 1.161.17 25’ - 2’ 4.214.19 VB12.6512.6 a0a0 5.4665.431 EbEb 4.954.75 B0.99 Present:

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 6 Examples GGAHybridExpt. a) Stable (0)C 3v C 1h E(+/0)0.660.94 b) 0.99 E(0/-)0.550.66 b) 0.75 C 1h C 3v a) Watkins et al. PRB 12, 5824 (1975); 36, 1094 (1987) b) GGA E BE corrected with gap level positions in Hybrid. GGA Hybr. GGA Hybr. LDAHybridGW 3 V Si occupationa(1), e(1)e(2), a(0) E( 3 V Si ) - E(V C +C si )2.251.19 e a a VB CB e a a LDA O i diffusion in Silicon Si 64 ; 4 4 4; 21G* (0.12HF + 0.88PBE) Neutral BI in Silicon Si 64 ; 2 2 2; 21G* (0.12HF + 0.88PBE) V Si metastability in 4H-SiC Si 64 C 64 ; 4 4 4; 21G* (0.2HF + 0.8PZ) Experiment (270-700 °C): 2.53 eV Stavola et al., APL. 42, 73 (1983); Takeno et al., JAP 84, 3113 (1998). 2.62 2.30 E tot (O y )-[E tot (O i )+ZPE] e D -e V ) 0.64 0.36 OiOi OiOi OYOY GGA Hybrid

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 7 An approximate correction Implication of the previous examples: the error in energy differences between two configurations is related the error in the gap level position! Let us introduce a correction! LDA or GGA HF or GW Total Energy Difference LDA“Marker”HF-corrected LDA HybridExptl. a) Si: H BC (+/0)+0.54+0.83+0.98+0.94 Si: BI (+/0)+0.66+0.94+0.99 a) K. B. Nielsen et al., PR B 65, 075205 (2002). b) Watkins et al. PRB 12, 5824 (1975); 36, 1094 (1987) Seems to work well for charge transition levels! SMALL CHANGES IN: GOOD CANCELLATION

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Peter De á k deak@bccms.uni-bremen.de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 8 Cancellation when the configuration changes?? Si 64 Si 64 C 64 a - bH BC (0) – H AB (0)O Y (0) –O i (0) 3 V Si (0) – V C +C Si (0) -0.04+0.51+1.08 +0.05-0.21-1.83 +0.01+0.30-0.75 +0.19+0.27-1.89 CONCLUSIONS: 1.LDA or GGA give rise to an error in the band energy E BE (“gap error” ), which is defect dependent. 2.The error in E BE is not – as a rule – compensated in the expression of the total energy E tot ! 3.Calculated energy differences between different charge states are not – as a rule – correct! 5.Total energy differences may be seriously wrong, for defects with different kinds of bonding configuration and levels in the gap. The ground state may not be predicted correctly! At least checks with hybrid functionals are recommended! 6.There are catastrophe cases (e.g., TiO 2 :V O )! 4.If the bonding configuration does not change much, correction of the gap level in E BE is sufficient, but only then!

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