Photoelectronic properties of chalcopyrites for photovoltaic conversion:

Transcription

1 Photoelectronic properties of chalcopyrites for photovoltaic conversion: self-consistent GW calculations Silvana Botti 1 LSI, CNRS-CEA-École Polytechnique, Palaiseau, France 2 LPMCN, CNRS-Université Lyon 1, France 3 European Theoretical Spectroscopy Facility September 13, rd International Workshop Time dependent Density-Functional Theory: Prospects and Applications Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 1 / 33

2 Collaborators Lun Mei Huang Pa r Olsson Christophe Domain Jean-Franc ois Guillemoles Julien Vidal Matteo Gatti Lucia Reining LSI CNRS-E cole Polytechnique Silvana Botti (ETSF) IRDEP CNRS-EDF Self-consistent GW for CIS TDDFT Workshop 2 / 33

3 Outline 1 Photovoltaic materials 2 Beyond Standard DFT 3 Beyond Standard GW Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 3 / 33

4 Photovoltaic materials Present state of photovoltaic efficiency from National Renewable Energy Laboratory (USA) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 4 / 33

5 CIS solar cell Photovoltaic materials Devices have to fulfill 2 functions: Photogeneration of electron-hole pairs Separation of charge carriers to generate a current Structure: Molybdenum back contact CIS layer (p-type layer) CdS layer (n-type layer) ZnO:Al transparent contact Efficiency = 13 % Würth Elektronik GmbH & Co. Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 5 / 33

6 CIS properties Photovoltaic materials CuIn(S,Se) 2 are among the best photovoltaic absorber materials: high optical absorption thin layer films optimal photovoltaic gap (record efficiency 19.9 %) self-doping with native defects p-n junctions electrical tolerance to large off-stoichiometries: not yet understood benign character of defects: not yet understood Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 6 / 33

7 Photovoltaic materials Density functional theory DFT in its standard form is a ground state theory Structural parameters: lattice parameters, internal distortions are OK, even in LDA or GGA Formation energies for defects calculated from total energies are reliable Kohn-Sham energies are not meant to reproduce quasiparticle band structures: often one obtains good band dispersions but band gaps are systematically underestimated Kohn-Sham DOS is not meant to reproduce photoemission Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 7 / 33

8 Photovoltaic materials State of the art for CuIn(S,Se) 2 First ab initio calculation for chalcopyrite Jaffe et al., PRB 28,10 (1983) Formation energies of intrinsic defects and defect levels in the gap Zhang et al., PRB 57, 9642 (1998) Correction of the bandgap DFT+U E v = ev Lany et al., PRB (2005) Metastability caused by the vacancy complex V Se -V Cu Lany et al., JAP, 100, (2006) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 8 / 33

9 Photovoltaic materials Modeling photovoltaic materials Objectives Predict accurate values for fundamental opto-electronical properties of materials Deal with complex materials (large unit cells, defects) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 9 / 33

10 Photovoltaic materials LDA Kohn-Sham energy gaps calculated gap (ev) HgTe InSb,P,InAs InN,Ge,GaSb,CdO Si InP,GaAs,CdTe,AlSb Se,Cu2O AlAs,GaP,SiC,AlP,CdS ZnSe,CuBr ZnO,GaN,ZnS diamond SrO AlN :LDA CaO MgO experimental gap (ev) van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 10 / 33

11 Photovoltaic materials LDA Kohn-Sham energy gaps for CIS CuInS 2 DFT-LDA exp. E g In-S S s band In 4 d band CuInSe 2 DFT-LDA exp. E g In-Se Se s band In 4 d band Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 11 / 33

12 Beyond Standard DFT Beyond standard DFT For photovoltaic applications we are interested in evaluating quasiparticle band gap optical band gap defect energy levels optical absorption spectra All these quantities require going beyond standard DFT Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 12 / 33

13 Solution Beyond Standard DFT Π0 In the many-body framework, we know how to solve these problems: GW for quasi-particle properties Bethe-Salpeter equation for the inclusion of electron-hole interaction The first step can be substantially more complicated than the second, so in the following we will focus on GW Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 13 / 33

14 Beyond Standard DFT Hedin s equations W W = v + vpw Σ = GWΓ P Σ P = GGΓ G=G 0 +G 0 Σ G Γ G Γ=1+(δΣ/δG)GGΓ L. Hedin, Phys. Rev. 139 (1965). Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 14 / 33

15 Beyond Standard DFT Self-energy and screened interaction Self-energy: nonlocal, non-hermitian, frequency dependent operator It allows to obtain the Green s function G once that G 0 is known Hartree-Fock Σ x (r 1, r 2 ) = ig(r 1, r 2, t, t + )v(r 1, r 2 ) GW Σ(r 1, r 2, t 1 t 2 ) = ig(r 1, r 2, t 1 t 2 )W (r 1, r 2, t 2 t 1 ) W = ɛ 1 v: screened potential (much weaker than v!) Ingredients: KS Green s function G 0, and RPA dielectric matrix ɛ 1 G,G (q, ω) L. Hedin, Phys. Rev. 139 (1965) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 15 / 33

16 Beyond Standard DFT Standard one-shot GW Kohn-Sham equation: H 0 (r)ϕ KS (r) + v xc (r) ϕ KS (r) = ε KS ϕ KS (r) Quasiparticle equation: H 0 (r)φ QP (r) + dr Σ ( r, r ) (, ω = E QP φqp r ) = E QP φ QP (r) Quasiparticle energies 1st order perturbative correction with Σ = igw : E QP ε KS = ϕ KS Σ v xc ϕ KS Basic assumption: φ QP ϕ KS Hybersten and Louie, PRB 34 (1986); Godby, Schlüter and Sham, PRB 37 (1988) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 16 / 33

17 Beyond Standard DFT Energy gap within standard one-shot GW calculated gap (ev) HgTe InSb,P,InAs InN,Ge,GaSb,CdO Si InP,GaAs,CdTe,AlSb Se,Cu2O AlAs,GaP,SiC,AlP,CdS ZnSe,CuBr ZnO,GaN,ZnS diamond SrO AlN CaO MgO 0 :LDA :GW(LDA) experimental gap (ev) van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 17 / 33

18 Beyond Standard DFT Quasiparticle energies within G 0 W 0 for CIS CuInS 2 DFT-LDA G 0 W 0 exp. E g In-S S s band In 4 d band CuInSe 2 DFT-LDA G 0 W 0 exp. E g In-Se Se s band In 4 d band Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 18 / 33

19 Beyond Standard GW Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) Semi-empirical hybrid functionals (e.g. Fuchs et al. 2007) Self-consistent approaches: sccohsex scheme (Hedin 1965, Bruneval et al. 2005) GWscQP scheme (Faleev et al. 2004) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 19 / 33

20 Beyond Standard GW Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) Semi-empirical hybrid functionals (e.g. Fuchs et al. 2007) Self-consistent approaches: sccohsex scheme (Hedin 1965, Bruneval et al. 2005) GWscQP scheme (Faleev et al. 2004) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 19 / 33

21 Beyond Standard GW Self-consistent COHSEX Coulomb hole: Σ COH (r 1, r 2 ) = 1 2 δ(r 1 r 2 )[W (r 1, r 2, ω = 0) v(r 1, r 2 )] Screened Exchange: Σ SEX (r 1, r 2 ) = i θ(µ E i )φ i (r 1 )φ i (r 2)W (r 1, r 2, ω = 0) The COHSEX self-energy is static and Hermitian Self-consistency can be done either on energies alone or on both energies and wavefunctions Representation of WFs on a restricted LDA basis set Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 20 / 33

22 Beyond Standard GW Self-consistent GW à la Faleev Make self-energy Hermitian and static ki Σ kj = 1 4 ( ki Σ(εkj ) kj + kj Σ(ε kj ) ki + ki Σ(ε ki ) kj + kj Σ(ε ki ) ki ) ki and ε ki are self-consistent eigensolutions of the iterative procedure Representation of WFs on a restricted LDA basis set Requires sums over empty states Faleev, van Schilfgaarde, and Kotani, PRL Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 21 / 33

23 Beyond Standard GW Self-consistent COHSEX Advantages of COHSEX: Old approximation physically motivated: accounts for Coulomb-hole and screened-exchange Computationally inexpensive : static; only sums over occupied states sc-cohsex wave-functions very similar to sc-gw Disadvantages of COHSEX: Dynamical correlations are missing Quasiparticle gaps are better (10-20% higher than experiment), but still not OK One-shot GW on top of sc-cohsex corrects the energy gap! Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 22 / 33

24 Beyond Standard GW Energy gap within sc GW QPscGW gap (ev) HgTe InSb,InAs InN,GaSb InP,GaAs,CdTe Cu2O ZnTe,CdS ZnSe,CuBr ZnO,GaN ZnS Si Ge,CdO P,Te AlN MgO CaO SrO diamond AlAs,GaP,SiC,AlP AlSb,Se experimental gap (ev) van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 23 / 33

25 Beyond Standard GW Quasiparticle energies within sc-gw for CIS CuInS 2 DFT-LDA G 0 W 0 sc-gw exp. E g In-S S s band In 4 d band CuInSe 2 DFT-LDA G 0 W 0 sc-gw exp. E g (+0.2) In-Se Se s band In 4 d band sc-gw is here sc-cohsex+g 0 W 0 Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 24 / 33

26 Results for CIS Beyond Standard GW Self-consistency in the energies suffices to correct the gap Self-consistency in the wave-functions is necessary to correct deeper states Spin-orbit coupling is important for Se compound Can be added perturbatively within DFT Experiment: 0.2 ev Calculation: 0.16 ev Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 25 / 33

27 Beyond Standard GW Example: Insulating phase of Vanadium Oxide LDA valence WF sc-gw: variation of WF metal E g =0.65 ev The variation of the wave-functions is essential to open up the gap (metal both in LDA and G 0 W 0!) Gatti et al., PRL 99 (2007) Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 26 / 33

28 Beyond Standard GW Quasi-particle corrections for CuInS 2 E GW -E DFT (ev) 0 Bandgap region -2 In-S bond S 3s -4 Cu 3d S 3p In 4d T Conduction bands Cu 3d S 3p Γ N E DFT (ev) Corrections depend on the character of the band Models for quasi-particle corrections for defects Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 27 / 33

29 Beyond Standard GW K-point dependence of the correction Corrections are independent of the k-point (within 0.1 ev) Effective masses remain unchanged Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 28 / 33

30 Beyond Standard GW Quasi-particle corrections for CuInSe 2 2 E GW -E DFT [ev] DOS [a.u.] KS energies [ev] blue: G 0 W 0 black: sc-cohsex red: sc-cohsex + G 0 W 0 Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 29 / 33

31 Beyond Standard GW Quasi-particle corrections for CuInS 2 2 E GW -E LDA [ev] DOS [a.u] KS Energies [ev] blue: G 0 W 0 black: sc-cohsex red: sc-cohsex + G 0 W 0 Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 30 / 33

32 Defects Beyond Standard GW a) Perfect crystal b) V Cu c) V Se d) 2V Cu In Cu (a) (b) DOS (c) (d) Energy (ev) Gap in the valence disappears in the presence of defects Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 31 / 33

33 Beyond Standard GW Conclusions and perspectives Methods that go beyond ground-state DFT are by now well established GW and BSE Self-consistency is absolutely necessary for d-electrons Self-consistent GW gives a very good description of quasi-particle states Self-consistent COHSEX good starting point for one-shot GW In all cases we studied this proved to be at the level of scgw Much more friendly from the computational point of view In progress Defects Absorption spectra from the Bethe-Salpeter equation Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 32 / 33

34 Beyond Standard GW Thanks!!! Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 33 / 33