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

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1 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

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

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

4 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

5 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 LDA+GW pseudopotential LDA+GW 1,2 pseudopotential LDA+GW 3 all-electron Experiment 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

6 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 LDA+GW pseudopotential LDA+GW 1,2 pseudopotential LDA+GW 3 all-electron Experiment 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

7 Many-Body Perturbation Theory Many-Body Green s Function Theory non-interacting electrons: Density-Functional Theory (DFT) ] G 1 DFT (r, r ; ɛ) = [ɛ v ext(r) v H (r) v xc (r) δ(r r ) interacting electrons: self-energy (Σ xc ) G 1 (r, r ; ɛ) = ] [ɛ 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

8 Many-Body Perturbation Theory Many-Body Green s Function Theory non-interacting electrons: Density-Functional Theory (DFT) ] G 1 DFT (r, r ; ɛ) = [ɛ v ext(r) v H (r) v xc (r) δ(r r ) interacting electrons: self-energy (Σ xc ) G 1 (r, r ; ɛ) = ] [ɛ 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

9 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

10 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

11 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

12 Computational Approach DFT: Kohn-Sham equation: [ ] 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

13 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 [ ] 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

14 Computational Approach DFT: Kohn-Sham equation: [ ] 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

15 Computational Approach DFT: Kohn-Sham equation: [ ] 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

16 Details of the Calculation plane-wave DFT program: SFHIngX ( 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 ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk7 Univers / 21

17 Details of the Calculation plane-wave DFT program: SFHIngX ( 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

18 Details of the Calculation plane-wave DFT program: SFHIngX ( 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 Ry 45 Ry 40 Ry ZnO Ry 35 Ry 56 Ry ZnS Ry 35 Ry 40 Ry CdS Ry 30 Ry 24 Ry ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005 Yarmouk7 Univers / 21

19 OEPx versus LDA: Zn and Ga Atoms Energy [ev] p 4s 4s 3d 3d 4p -I(exp)=-9.39 ev LDA OEPx(cLDA) -e 2 /r Zn Energy [ev] d 3d 4s 4p 4p 4s -I(exp)=-6.00 ev LDA OEPx(cLDA) -e 2 /r Ga r [a.u.] 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

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

21 OEPx versus LDA: ZnS compared to GaN electrons/bohr Zn n i (r) band 1 band 2-6 band n(r) S Zn S OEPx(cLDA) LDA Difference x10 x10 electrons/bohr Ga n i (r) band 1-6 band 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

22 Band Gaps and d-electrons: ZnS ev ZnS - Band Gap ev ZnS - d-electron binding energy 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

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

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

25 Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO rd shell 2, OEPx+GW Experiment 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

26 Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO rd shell 2, OEPx+GW Experiment d-electron binding energy [ev] OEPx+GW Experiment 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

27 Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO rd shell 2, OEPx+GW Experiment d-electron binding energy [ev] LMTO rd shell 2, OEPx+GW Experiment 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

28 Band Gaps and d-electrons GW ZnO ZnS CdS GaN E gap [ev] LMTO rd shell 2, OEPx+GW Experiment d-electron binding energy [ev] LMTO rd shell 2, SAT OEPx+GW Experiment 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

29 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

30 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

31 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

32 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

33 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

34 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 } ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

35 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 LDA CBM OEPx(cLDA) GaN CBM LDA OEPx(cLDA) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

36 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 OEPx(cLDA) VBM LDA OEPx(cLDA) GaN CBM VBM LDA OEPx(cLDA) LDA OEPx(cLDA) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

37 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 OEPx(cLDA) VBM LDA OEPx(cLDA) d-state LDA OEPx(cLDA) CBM LDA OEPx(cLDA) VBM LDA OEPx(cLDA) d-state LDA OEPx(cLDA) ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 14 Univers / 21

38 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: ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 15 Univers / 21

39 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 ) Preprint available at: ( Fritz-Haber-Institut Combining GW with der Max-Planck-Gesellschaft, OEPx Berlin - Germany OEP 2005Yarmouk 16 Univers / 21

40 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

41 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

42 Bandstructure of ZnO ZnO, zincblend ZnO, zincblend LDA+GW LDA 2 EXX+GW EXX Energy [ev] 0-2 Energy [ev] 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

43 Bandstructure of ZnS Energy [ev] ZnS LDA+GW LDA Energy [ev] 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

44 Bandstructure of GaN Energy [ev] GaN LDA+GW LDA -18 L Γ X W K Γ Energy [ev] 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

45 Quasiparticle Shifts Quasiparticle Shift [ev] ZnS OEPx(cLDA)+GW LDA+GW GaN OEPx(cLDA)+GW LDA+GW Quasiparticle Shift [ev] ε DFT -ε VBM [ev] 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