Challenges for ab initio defect modeling Peter Dek

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 Peter Deák deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 1

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

“State of the art” • Supercell: Nr. of Atoms 64 128 216 256 Nr. of k points 43 23 1 1 • Plane Waves (with UPP or PAW ) up to ~240 e. V. • DFT-GGA: (PBE functional) • commercial or public domain „turn-key“ package WHAT COULD POSSIBLY GO WRONG? BAND GAP & GAP STATES! e. C e. V Shishkin&Kresse, PRB 75, 235102 (2007) Peter Deák “the scissor” e. C ? ? e. V Scissor works only for defects in the high electron density region of the perfect crystal. Deák et al. . PRB 75, 153204 (2007) deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 3

Popular misapprehensions 1. 2. GGA is always successful in describing the ground state of a system. Internal ionization energies (charge transition levels) of defects can be calculated accurately as difference between total energies. Considering vertical transitions (no relaxation of ions) as in optical absorption experiments: 0 Kohn-Sham levels (w. gap error) e. C Eg Total energies (w. o. gap error) e. D e. V Problem of charged supercells can be handled by the Makov-Payne correction [PRB 51, 4014 (1995)]. ASSUMPTIONS 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). Si. C: VSi The total energy is not affected by the “gap error”! Cancellation? ? Peter Deák deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 4

Hybrid functionals as etalon 1. 2. 3. M. Städele et al. , PRB 59, 10031 (1999): “most of the gap error disappears when using exact exchange in DFT”. A. D. Becke, JCP 107, 8554 (1997): “mixing HF-exchange to DFT improves calculated molecular properties” J. Muscat et al, Chem. Phys. Lett. 342, 397 (2001): “in solids the gap improves as well”. 4. M. Marsman et al. , J. Phys. : Condens. Matter. 20, 064201 (2008): LATTICE CONSTANT BAND GAP Present: BULK MODULUS 0. 12 HF + 0. 88 PBE Exptl. Eg 1. 16 1. 17 G 25’-G 2’ 4. 21 4. 19 VB 12. 65 12. 6 a 0 5. 466 5. 431 Eb 4. 95 4. 75 B 0. 99 Defect levels COHESIVE ENERGY Peter Deák deak@bccms. uni-bremen. de e. D-e. V LDA G 0 W 0 Hybrid Si: HBC (0) +0. 61 +1. 05 +1. 08 Si: HAB (-) -0. 07 +0. 10 +0. 05 4 H-Si. C: HAB (-) +0. 50 +0. 62 +0. 54 3 C-Si. C: BSi+2 Ci (+) -0. 12 +0. 04 +0. 10 3 C-Si. C: BSi+2 Ci (-) +0. 18 +0. 26 +0. 29 Challenges for ab initio defect modeling. EMRS Symposium I, 2008 5

Oi diffusion in Silicon Examples Neutral BI in Silicon Si 64; 2 2 2; 21 G* (0. 12 HF + 0. 88 PBE) Si 64; 4 4 4; 21 G* (0. 12 HF + 0. 88 PBE) OY Oi Oi Hybr. GGA C 1 h e. D-e. V) Etot(Oy)-[Etot(Oi)+ZPE] 2. 62 2. 30 GGA Hybrid GGA 0. 64 0. 36 Experiment (270 -700 °C): 2. 53 e. V Stavola et al. , APL. 42, 73 (1983); Takeno et al. , JAP 84, 3113 (1998). C 3 v GGA Hybrid Expt. a) Stable (0) C 3 v C 1 h E(+/0) 0. 66 0. 94 b) 0. 99 E(0/-) 0. 55 0. 66 b) 0. 75 a) Watkins et al. PRB 12, 5824 (1975); 36, 1094 (1987) b) GGA EBE corrected with gap level positions in Hybrid. VSi metastability in 4 H-Si. C Si 64 C 64; 4 4 4; 21 G* (0. 2 HF + 0. 8 PZ) CB e VB a a LDA e a 3 V Si occupation E(3 VSi) - E(VC+Csi) LDA Hybrid GW a(1), e(1) e(2), a(0) 2. 25 1. 19 a Peter Deák deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 6

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 Seems to work well for charge transition levels! Total Energy Difference LDA “Marker” HF-corrected LDA Hybrid Exptl. a) Si: HBC(+/0) +0. 54 +0. 83 +0. 98 +0. 94 Si: BI (+/0) +0. 66 a) K. B. Nielsen et al. , PR B 65, 075205 (2002). b) Watkins GOOD CANCELLATION +0. 94 +0. 99 et al. PRB 12, 5824 (1975); 36, 1094 (1987) SMALL CHANGES IN: Peter Deák deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 7

Cancellation when the configuration changes? ? Si 64 a-b Si 64 C 64 HBC(0) – HAB(0) OY(0) –Oi(0) 3 V Si(0) – VC+CSi(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 EBE (“gap error”), which is defect dependent. 2. The error in EBE is not – as a rule – compensated in the expression of the total energy Etot ! 3. Calculated energy differences between different charge states are not – as a rule – correct! 4. If the bonding configuration does not change much, correction of the gap level in EBE is sufficient, but only then! 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! 6. There are catastrophe cases (e. g. , Ti. O 2: VO)! Peter Deák At least checks with hybrid functionals are recommended! deak@bccms. uni-bremen. de Challenges for ab initio defect modeling. EMRS Symposium I, 2008 8