Combining quasiparticle energy calculations with exact-exchange density-functional theory

Combining quasiparticle energy calculations with exact-exchange density-functional theory Patrick Rinke 1, Abdallah Qteish 1,2, Jörg Neugebauer 1,3,4, Christoph Freysoldt 1 and Matthias Scheffler 1 1 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany 2 Yarmouk University, Irbid - Jordan 3 University of Paderborn, Germany 4 Max-Planck-Institut für Eisenforschung, Düsseldorf - Germany OEP Workshop, 2005 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk1 Univers / 21

Medium Band Gap Materials II-VI: ZnO, ZnS, CdS III-V: GaN are medium band gap semiconductors: E gap = 2.5-4.0 ev interesting for applications in optical industry have semicore d-states with low binding energies: E d = 9.0-18.0 ev ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk2 Univers / 21

Medium Band Gap Materials II-VI: ZnO, ZnS, CdS III-V: GaN are medium band gap semiconductors: E gap = 2.5-4.0 ev interesting for applications in optical industry have semicore d-states with low binding energies: E d = 9.0-18.0 ev Schematic band diagram: ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk2 Univers / 21

Ab initio Bandstructures (Inverse) photoemission probes: excited hole and electron states quasiparticles GW approximation: method of choice for quasiparticle excitations in solids typically applied as perturbation to DFT in LDA ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk3 Univers / 21

Ab initio Bandstructures (Inverse) photoemission probes: excited hole and electron states quasiparticles GW approximation: method of choice for quasiparticle excitations in solids typically applied as perturbation to DFT in LDA E gap [ev] approach ZnO ZnS CdS GaN DFT-LDA pseudopotential 0.51 1.76 0.81 1.52 LDA+GW pseudopotential 1.36 2.59 1.60 2.39 LDA+GW 1,2 pseudopotential 3.50 3.45 2.88 LDA+GW 3 all-electron 3.24 3.03 Experiment 3.44 3.80 2.48 3.30 1 Rohlfing, Krüger, Pollmann PRB 57 (1998) 2 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) 3 Kotani, van Schilfgaarde SSC 121 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk3 Univers / 21

Ab initio Bandstructures (Inverse) photoemission probes: excited hole and electron states quasiparticles GW approximation: method of choice for quasiparticle excitations in solids typically applied as perturbation to DFT in LDA alternative: apply as perturbation to DFT in exact-exchange E gap [ev] approach ZnO ZnS CdS GaN DFT-LDA pseudopotential 0.51 1.76 0.81 1.52 LDA+GW pseudopotential 1.36 2.59 1.60 2.39 LDA+GW 1,2 pseudopotential 3.50 3.45 2.88 LDA+GW 3 all-electron 3.24 3.03 Experiment 3.44 3.80 2.48 3.30 1 Rohlfing, Krüger, Pollmann PRB 57 (1998) 2 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) 3 Kotani, van Schilfgaarde SSC 121 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk3 Univers / 21

Many-Body Perturbation Theory Many-Body Green s Function Theory non-interacting electrons: Density-Functional Theory (DFT) ] G 1 DFT (r, r ; ɛ) = [ɛ + 2 2 v ext(r) v H (r) v xc (r) δ(r r ) interacting electrons: self-energy (Σ xc ) G 1 (r, r ; ɛ) = ] [ɛ + 2 2 v ext(r) v H (r) δ(r r ) Σ xc (r, r ; ɛ) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk4 Univers / 21

Many-Body Perturbation Theory Many-Body Green s Function Theory non-interacting electrons: Density-Functional Theory (DFT) ] G 1 DFT (r, r ; ɛ) = [ɛ + 2 2 v ext(r) v H (r) v xc (r) δ(r r ) interacting electrons: self-energy (Σ xc ) G 1 (r, r ; ɛ) = ] [ɛ + 2 2 v ext(r) v H (r) δ(r r ) Σ xc (r, r ; ɛ) Perturbation Theory adiabatic connection between non-interacting and interacting system Dyson equation: G = G DFT + G DFT [Σ xc v xc ] G ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk4 Univers / 21

GW Approximation Density-Functional Theory exact exchange-correlation potential v xc unknown approximate v xc approximate G DFT ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk5 Univers / 21

GW Approximation Density-Functional Theory exact exchange-correlation potential v xc unknown approximate v xc approximate G DFT GW approximation exact self-energy Σ xc given by Hedin s equations (not tractable) GW approximation: Σ xc = igw approximate G ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk5 Univers / 21

GW Approximation Density-Functional Theory exact exchange-correlation potential v xc unknown approximate v xc approximate G DFT GW approximation exact self-energy Σ xc given by Hedin s equations (not tractable) GW approximation: Σ xc = igw approximate G non self-consistent GW : G 0 = G DFT χ 0 (1, 2) = ig 0 (1, 2)G 0 (2, 1 + ) W 0 (1, 2) = v(1, 2) + v(1, 3)χ 0 (3, 4)W 0 (4, 2)d(3, 4) Σ GW 0 (1, 2) = ig 0 (1, 2)W 0 (1, 2) self-energy depends on ground state (G 0, v xc ) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk5 Univers / 21

Computational Approach DFT: Kohn-Sham equation: [ ] 2 2 + v ext(r) + v H (r) + v xc (r) φ nk (r) = ɛ nk φ nk (r) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk6 Univers / 21

Computational Approach DFT: Kohn-Sham equation: Local-density approximation (LDA) local potential v xc (r) with parametrised density dependence fast to compute not self-interaction free eigenvalues poor excitations [ ] 2 2 + v ext(r) + v H (r) + v xc (r) φ nk (r) = ɛ nk φ nk (r) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk6 Univers / 21

Computational Approach DFT: Kohn-Sham equation: [ ] 2 2 + v ext(r) + v H (r) + v xc (r) φ nk (r) = ɛ nk φ nk (r) Local-density approximation (LDA) local potential v xc (r) with parametrised density dependence fast to compute not self-interaction free eigenvalues poor excitations Exact-exchange (OEPx) best local groundstate potential v x (r) to non-local Σ x (r, r ) OEP formalism computationally demanding self-interaction free eigenvalues reasonable excitations ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk6 Univers / 21

Computational Approach DFT: Kohn-Sham equation: [ ] 2 2 + v ext(r) + v H (r) + v xc (r) φ nk (r) = ɛ nk φ nk (r) Local-density approximation (LDA) local potential v xc (r) with parametrised density dependence fast to compute not self-interaction free eigenvalues poor excitations Exact-exchange (OEPx) best local groundstate potential v x (r) to non-local Σ x (r, r ) OEP formalism computationally demanding self-interaction free eigenvalues reasonable excitations GW : Quasiparticle equation: φ qp nk (r) φdft nk (r) ɛ qp nk = ɛdft nk What is the better starting point for GW calculation? + φ nk Σ GW xc (ɛ qp nk ) v xc µ φ nk ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk6 Univers / 21

Details of the Calculation plane-wave DFT program: SFHIngX (http://www.sfhingx.de) all OEPx calculations include LDA correlation LDA and OEPx pseudopotentials with 3d in valence (OEPx PSP: Moukara, Städele, Görling et al. J. Phys. Cond. Mat. 12 (2000)) only zinc-blende structures GW space-time code: GWST(Steinbeck, Godby et al. CPC 117 (1999), CPC 125 (2000)) time/frequency cutoff 12 Ry with 18 points per axis ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk7 Univers / 21

Details of the Calculation plane-wave DFT program: SFHIngX (http://www.sfhingx.de) all OEPx calculations include LDA correlation LDA and OEPx pseudopotentials with 3d in valence (OEPx PSP: Moukara, Städele, Görling et al. J. Phys. Cond. Mat. 12 (2000)) only zinc-blende structures GW space-time code: GWST(Steinbeck, Godby et al. CPC 117 (1999), CPC 125 (2000)) time/frequency cutoff 12 Ry with 18 points per axis a ZB E cut χ cut b cut Å GW,OEPx OEPx GW GaN 4.50 70 Ry 45 Ry 40 Ry ZnO 4.62 60 Ry 35 Ry 56 Ry ZnS 5.40 60 Ry 35 Ry 40 Ry CdS 5.82 50 Ry 30 Ry 24 Ry ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk7 Univers / 21

OEPx versus LDA: Zn and Ga Atoms Energy [ev] 0.0-5.0-10.0-15.0-20.0-25.0 4p 4s 4s 3d 3d 4p -I(exp)=-9.39 ev LDA OEPx(cLDA) -e 2 /r Zn Energy [ev] 0.0-5.0-10.0-15.0-20.0-25.0 3d 3d 4s 4p 4p 4s -I(exp)=-6.00 ev LDA OEPx(cLDA) -e 2 /r Ga 0 2 4 6 8 10 12 r [a.u.] 0 2 4 6 8 10 12 r [a.u.] OEPx PSP according to: Moukara, Städele, Majewski, Vogl and Görling, J. Phys. Cond. Mat. 12 (2000) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk8 Univers / 21

OEPx versus LDA: ZnS valence bands (band 7-9) [100] [011] Zn S 1.0e-02-1.0e-02-3.0e-02-5.0e-02-7.0e-02 ZnS [111] [111] [100] 10.4e-02 d-bands (band 2-6) [011] Zn S 7.6e-02 4.8e-02 2.0e-02-0.8e-02 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk9 Univers / 21

OEPx versus LDA: ZnS compared to GaN electrons/bohr 3 0.12 0.08 0.04 0.00-0.04 Zn n i (r) band 1 band 2-6 band 7-9 2.0 1.5 1.0 0.5 0.0 n(r) S Zn S OEPx(cLDA) LDA Difference x10 x10 electrons/bohr 3 0.12 0.08 0.04 0.00-0.04 Ga n i (r) band 1-6 band 7-9 2.00 1.50 1.00 0.50 0.00 n(r) N Ga N OEPx(cLDA) LDA Difference x5 x5 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 10 Univers / 21

Band Gaps and d-electrons: ZnS ev ZnS - Band Gap ev ZnS - d-electron binding energy 3.70 3.80 8.97 3.08 7.05 7.08 2.59 2.65 6.32 2.19 5.33 5.02 1.76 4.30 LDA Pseudopotentials OEPx Pseudopotentials LDA Pseudopotentials OEPx Pseudopotentials LDA OEPx LDA + GW OEPx + GW OEPx OEPx + Exp GW LDA OEPx LDA + GW OEPx + GW OEPx OEPx + Exp GW ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 11 Univers / 21

Band Gaps and d-electrons ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] OEPx+GW 3.1 3.7 2.4 3.1 Experiment 3.4 3.8 2.5 3.3 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO 1 3.2 3.0 3 rd shell 2,3 3.5 2.5 2.9 OEPx+GW 3.1 3.7 2.4 3.1 Experiment 3.4 3.8 2.5 3.3 1 Kotani, van Schilfgaarde SSC 121 (2002) 2 Rohlfing, Krüger, Pollmann PRB 57 (1998) 3 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO 1 3.2 3.0 3 rd shell 2,3 3.5 2.5 2.9 OEPx+GW 3.1 3.7 2.4 3.1 Experiment 3.4 3.8 2.5 3.3 d-electron binding energy [ev] OEPx+GW 6.9 7.1 7.8 16.2 Experiment 9.0 9.0 9.5 17.7 1 Kotani, van Schilfgaarde SSC 121 (2002) 2 Rohlfing, Krüger, Pollmann PRB 57 (1998) 3 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO 1 3.2 3.0 3 rd shell 2,3 3.5 2.5 2.9 OEPx+GW 3.1 3.7 2.4 3.1 Experiment 3.4 3.8 2.5 3.3 d-electron binding energy [ev] LMTO 1 7.1 8.2 16.4 3 rd shell 2,3 6.4 8.1 15.7 OEPx+GW 6.9 7.1 7.8 16.2 Experiment 9.0 9.0 9.5 17.7 1 Kotani, van Schilfgaarde SSC 121 (2002) 2 Rohlfing, Krüger, Pollmann PRB 57 (1998) 3 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO 1 3.2 3.0 3 rd shell 2,3 3.5 2.5 2.9 OEPx+GW 3.1 3.7 2.4 3.1 Experiment 3.4 3.8 2.5 3.3 d-electron binding energy [ev] LMTO 1 7.1 8.2 16.4 3 rd shell 2,3 6.4 8.1 15.7 SAT 2 7.9 9.1 17.3 OEPx+GW 6.9 7.1 7.8 16.2 Experiment 9.0 9.0 9.5 17.7 1 Kotani, van Schilfgaarde SSC 121 (2002) 2 Rohlfing, Krüger, Pollmann PRB 57 (1998) 3 Luo, Ismail-Beigi, Cohen, Louie PRB 66 (2002) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 12 Univers / 21

How to make pseudopotential GW work Systems with semicore states strong overlap of atomic wavefunctions in semicore shell strong core-valence exchange exchange has to be treated consistently recap: v x (r) is best local Σ x (r, r ) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 13 Univers / 21

How to make pseudopotential GW work Systems with semicore states strong overlap of atomic wavefunctions in semicore shell strong core-valence exchange exchange has to be treated consistently recap: v x (r) is best local Σ x (r, r ) DFT-LDA v LDA xc LDA+GW Σ xc exchange inconsistent v LDA xc v LDA xc ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 13 Univers / 21

How to make pseudopotential GW work Systems with semicore states strong overlap of atomic wavefunctions in semicore shell strong core-valence exchange exchange has to be treated consistently recap: v x (r) is best local Σ x (r, r ) DFT-LDA v LDA xc LDA+GW Σ xc exchange consistent v LDA xc v LDA xc ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 13 Univers / 21

How to make pseudopotential GW work Systems with semicore states strong overlap of atomic wavefunctions in semicore shell strong core-valence exchange exchange has to be treated consistently recap: v x (r) is best local Σ x (r, r ) DFT-OEPx OEPx+GW v OEPx x Σ xc v OEPx x v OEPx x ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 13 Univers / 21

Perturbation Operator and Self-Energy Quasiparticle Equation ɛ qp nk = ɛdft nk + φ nk Σ GW xc (ɛ qp nk ) v xc µ φ nk ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

Perturbation Operator and Self-Energy Quasiparticle Equation ɛ qp nk = ɛdft nk + φ nk Σ GW xc (ɛ qp nk ) v xc µ φ nk DFT influence: energies : ɛ DFT i band structure : { ɛ DFT nk, φ nk } wavefunctions : {φ i } ZnS State GS ɛ DFT Σ c Σ x v xc CBM LDA 1.75-4.33 4.68 OEPx(cLDA) 3.08-4.39 4.69 VBM LDA 0.00 1.62-2.04 OEPx(cLDA) 0.00 1.57-1.87 GaN CBM VBM LDA 1.65-4.27 4.72 OEPx(cLDA) 2.88-4.28 4.38 LDA 0.00 2.32-2.76 OEPx(cLDA) 0.00 2.32-2.43 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

Perturbation Operator and Self-Energy Quasiparticle Equation ɛ qp nk = ɛdft nk + φ nk Σ GW xc (ɛ qp nk ) v xc µ φ nk DFT influence: energies : ɛ DFT i band structure : { ɛ DFT nk, φ nk } wavefunctions : {φ i } ZnS GaN State GS ɛ DFT Σ c Σ x v xc CBM LDA 1.75-4.33 4.68 OEPx(cLDA) 3.08-4.39 4.69 VBM LDA 0.00 1.62-2.04 OEPx(cLDA) 0.00 1.57-1.87 d-state LDA -6.62 4.95-3.46 OEPx(cLDA) -7.29 5.27-5.71 CBM LDA 1.65-4.27 4.72 OEPx(cLDA) 2.88-4.28 4.38 VBM LDA 0.00 2.32-2.76 OEPx(cLDA) 0.00 2.32-2.43 d-state LDA -13.97 5.19-4.37 OEPx(cLDA) -14.89 5.50-6.85 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

Conclusions when the semicore d-electrons are considered as valence electrons: OEPx shows a considerable improvement over LDA and LDA+GW OEPx+GW band gaps are in very good agreement with experiment OEPx+GW d-electron binding energies are in agreement with previous GW calculations but still at variance with experiment OEPx constitutes the better starting point for GW consistency is paramount: OEPx pseudopotentials provide a much better description of core-valence exchange Preprint available at: http://arxiv.org/abs/cond-mat/0502404 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 15 Univers / 21

Outlook and Acknowledgements Outlook work in progress: more efficient implementation along the lines of Kümmel et al. PRL 90 (2003), PRB 68 (2003), PRL 93 (2004) all-electron OEPx calculations in progress in Graz (talk by Sangeeta Sharma) all-electron GW calculations in progress at the FHI in Berlin Acknowledgements Matthias Wahn, Philipp Eggert and Martin Fuchs A. Majewski and P. Vogl for their pseudopotential generator financial support by: Volkswagen Stiftung/Germany NANOQUANTA NoE (NMP4-CT-2004-500198) Preprint available at: http://arxiv.org/abs/cond-mat/0502404 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 16 Univers / 21

DFT in Exact-Exchange Formalism Optimised Effective Potential v x (r) best local potential to non-local exchange-self energy Σ x (r, r ) dr χ 0 (r, r )v x (r ) = dɛ dr dr G DFT (r, r ; ɛ)σ x (r, r )G DFT (r, r; ɛ). ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 17 Univers / 21

DFT in Exact-Exchange Formalism Optimised Effective Potential v x (r) best local potential to non-local exchange-self energy Σ x (r, r ) dr χ 0 (r, r )v x (r ) = dɛ dr dr G DFT (r, r ; ɛ)σ x (r, r )G DFT (r, r; ɛ). more familiar expression: occ v x (r) = dr i unocc s [ φ i Σ x φ s φ s(r )φ i (r ] ) + c.c. χ 1 0 ɛ i ɛ (r, r) s implemented in reciprocal space following: Görling PRB 53 (1996), Städele et al. PRL 79 (1997), PRB 59 (1999) work in progress: more efficient implementation along the lines of Kümmel et al. PRL 90 (2003), PRB 68 (2003), PRL 93 (2004) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 17 Univers / 21

Bandstructure of ZnO ZnO, zincblend ZnO, zincblend 6 6 4 4 2 LDA+GW LDA 2 EXX+GW EXX Energy [ev] 0-2 Energy [ev] 0-2 -4-4 -6-6 -8 L Γ X W K Γ -8 L Γ X W K Γ ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 18 Univers / 21

Bandstructure of ZnS Energy [ev] 8 6 4 2 0-2 -4-6 -8-10 -12 ZnS LDA+GW LDA Energy [ev] 8 6 4 2 0-2 -4-6 -8-10 -12 ZnS OEPx(cLDA)+GW OEPx(cLDA) L Γ X W K Γ L Γ X W K Γ ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 19 Univers / 21

Bandstructure of GaN Energy [ev] 10 8 6 4 2 0-2 -4-6 -8-10 -12-14 -16 GaN LDA+GW LDA -18 L Γ X W K Γ Energy [ev] 10 8 6 4 2 0-2 -4-6 -8-10 -12-14 -16 GaN OEPx(cLDA)+GW OEPx(cLDA) -18 L Γ X W K Γ ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk Univers / 21

Quasiparticle Shifts Quasiparticle Shift [ev] 1.0 0.5 0.0-0.5 ZnS OEPx(cLDA)+GW LDA+GW GaN OEPx(cLDA)+GW LDA+GW 1.0 0.5 0.0-0.5 Quasiparticle Shift [ev] -1.0-6 -5-4 -3-2 -1 0 ε DFT -ε VBM [ev] -1.0 0 1 2 3-6 -5-4 -3-2 -1 0 0 1 2 3 ε DFT -ε CBM ε DFT -ε VBM [ev] ε DFT -ε CBM ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 21 Univers / 21