Challenges for ab initio defect modeling

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Peter Deá[email protected]

Challenges for ab initio defect modeling. EMRS Symposium I, 2008

Challenges for ab initio defect modeling

Peter Deák, Bálint Aradi, and Thomas Frauenheim

Bremen Center for Computational Materials Science, University of BremenPOB 330440, 28334 Bremen, Germany

Adam GaliDept. Atomic Physics, Budapest University of Technology & Economics

H-1521 Budapest, Hungary

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How good is defect theory today?

Challenging some illusions!

Richard P MessmerGeorge D Watkins

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James W Corbett

control

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Shishkin&Kresse, PRB 75, 235102 (2007)

BAND GAP

“State of the art”

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• 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 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”

eV

eC

eV

eC

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SiC:VSi

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 misapprehensions1. 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.

€

I = Egexptl + ED

+ − ED0

( ) − Eperf+ − Eperf

0( )[ ]

0

Total energies (w.o. gap error)

ASSUMPTIONSProblem 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:

€

I = eC − eD = Eg − eD − eV( ) Kohn-Sham levels (w.gap error)

eD

eV

eC

Eg

The total energy is not affected by the “gap error”!The total energy is not affected by the “gap error”!

€

Etot = niei∑

EBE

1 2 3 + EdcCancellation??

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Hybrid functionals as etalon1. M. Städele et al., PRB 59, 10031 (1999): “most of the gap error disappears when using exact exchange in DFT”.

eD-eV LDA G0W0 Hybrid

Si: HBC (0) +0.61 +1.05 +1.08

Si: HAB (-) -0.07 +0.10 +0.05

4H-SiC: HAB (-) +0.50 +0.62 +0.54

3C-SiC: BSi+2Ci (+) -0.12 +0.04 +0.10

3C-SiC: BSi+2Ci (-) +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”.

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LATTICE CONSTANT

BULK MODULUS

COHESIVE ENERGY



BAND GAP

4. M. Marsman et al., J. Phys.: Condens. Matter. 20, 064201 (2008):

€

14 HF + 3

4 PBE = PBE0

hybrid exchange

0.12HF + 0.88PBE Exptl.

Eg 1.16 1.17

25’-2’ 4.21 4.19

VB 12.65 12.6

a0 5.466 5.431

Eb 4.95 4.75

B 0.99 0.99

Present:

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Examples

GGA Hybrid Expt.a)

Stable (0) C3v C1h C1h

E(+/0) 0.66 0.94b) 0.99

E(0/-) 0.55 0.66b) 0.75

C1hC3v

a) Watkins et al. PRB 12, 5824 (1975); 36, 1094 (1987)b) GGA EBE corrected with gap level positions in Hybrid.

GGA

Hybr.

GGAHybr.

LDA Hybrid GW

3VSi occupation a(1), e(1) e(2), a(0) e(2), a(0)

E(3VSi) - E(VC+Csi) 2.25 1.19eaa

VB

CB ea

a

LDA

Oi diffusion in SiliconSi64; 444; 21G* (0.12HF + 0.88PBE)

Neutral BI in SiliconSi64; 222; 21G* (0.12HF + 0.88PBE)

VSi metastability in 4H-SiCSi64C64; 444; 21G* (0.2HF + 0.8PZ)

Experiment (270-700 °C): 2.53 eVStavola et al., APL. 42, 73 (1983); Takeno et al., JAP 84, 3113 (1998).

2.622.30

Etot(Oy)-[Etot(Oi)+ZPE] eD-eV)

0.640.36

Oi OiOY

GGA

Hybrid

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An approximate correctionImplication 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!

€

Etot = nDεD + niε ii

VB

∑

EBE

1 2 4 3 4 4 + − 1

2 VH r( )ρ r( )dr∫ + EXC ρ r( )[ ]− VXC r( )ρ r( )dr∫{ }

Edc

1 2 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4

€

ΔεD = εDHForGW −εVBM

HForGW[ ]+ εD

GGAor LDA −εVBMGGAor LDA

[ ]

LDA or GGAHF or GW

Total Energy Difference

LDA “Marker” HF-corrected LDA

Hybrid Exptl.a)

Si: HBC(+/0) +0.54 +0.83 +0.98 +0.94 +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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Cancellation when the configuration changes??

Si64 Si64C64

a - b HBC(0) – HAB(0) OY(0) –Oi(0) 3VSi(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

€

Etot = nDεD + niε ii

VB

∑ + − 12 VH r( )ρ r( )dr∫ + EXC ρ r( )[ ]− VXC r( )ρ r( )dr∫{ } = EBE + Edc

CONCLUSIONS:1. LDA or GGA give rise to an error

in the band energy EBE (“gap

error”), which is defect dependent.

€

Δ EBEa − EBE

b( )

€

Δ Edca − Edc

b( )

€

Δ Etota − Etot

b( )

€

nDa eD

a + ΔeDa

( ) −

nDb eD

b + ΔeDb

( )

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!

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., TiO2:VO)!

4. If the bonding configuration does not change much, correction of the gap level in EBE is sufficient, but only then!

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Challenges for ab initio defect modeling