Electronic excitations in materials for solar cells

Transcription

1 Electronic excitations in materials for solar cells beyond standard density functional theory Silvana Botti 1 LSI, École Polytechnique-CNRS-CEA, Palaiseau, France 2 LPMCN, CNRS-Université Lyon 1, France 3 European Theoretical Spectroscopy Facility October 21, 2010 LASIM, Lyon Silvana Botti Electronic excitations & photovoltaics 1 / 34

2 Outline 1 Electronic excitations: beyond standard DFT? 2 Cu-based delafossite transparent conductive oxides 3 Optical spectra 4 Conclusions and perspectives Silvana Botti Electronic excitations & photovoltaics 2 / 34

3 Problems of standard DFT Modeling electronic excitations in complex systems Objectives Predict accurate values for fundamental opto-electronical properties (gap, absorption spectra, excitons,...) Simulate real materials (nanostructured systems, large unit cells, defects, doping, interfaces,...) Find a compromise between accuracy and computational effort Silvana Botti Electronic excitations & photovoltaics 3 / 34

4 Problems of standard DFT Modeling electronic excitations in complex systems Objectives Predict accurate values for fundamental opto-electronical properties (gap, absorption spectra, excitons,...) Simulate real materials (nanostructured systems, large unit cells, defects, doping, interfaces,...) Find a compromise between accuracy and computational effort Silvana Botti Electronic excitations & photovoltaics 3 / 34

5 Problems of standard DFT Density functional theory In literature standard computational approach for band structures: Kohn-Sham (KS) equations ] [ v ext (r) + v Hartree (r) + v xc (r) ϕ i (r) = ε i ϕ i (r) it is necessary to approximate v xc (r), Structural parameters and formation energies are usually good in LDA or GGA Kohn-Sham energies are not meant to reproduce quasiparticle band structures: one often obtains good band dispersions but band gaps are systematically underestimated and how to calculate optical absorption? Hohenberg&Kohn, PR 136, B864 (1964); Kohn&Sham, 140, A1133 (1965) Silvana Botti Electronic excitations & photovoltaics 4 / 34

6 Problems of standard DFT Density functional theory In literature standard computational approach for band structures: Kohn-Sham (KS) equations ] [ v ext (r) + v Hartree (r) + v xc (r) ϕ i (r) = ε i ϕ i (r) it is necessary to approximate v xc (r), Structural parameters and formation energies are usually good in LDA or GGA Kohn-Sham energies are not meant to reproduce quasiparticle band structures: one often obtains good band dispersions but band gaps are systematically underestimated and how to calculate optical absorption? Hohenberg&Kohn, PR 136, B864 (1964); Kohn&Sham, 140, A1133 (1965) Silvana Botti Electronic excitations & photovoltaics 4 / 34

7 Problems of standard DFT Excitation energies: photoemission Photoemission process: hν (E kin + φ) = E N 1,v E N,0 = ε v Silvana Botti Electronic excitations & photovoltaics 5 / 34

8 Problems of standard DFT Excitation energies: photoemission Inverse photoemission process: hν (E kin + φ) = E N,0 E N+1,c = ε c Silvana Botti Electronic excitations & photovoltaics 5 / 34

9 Problems of standard DFT Excitation energies: energy gap Photoemission gap: E gap = I A = min k,l ( EN 1,k + E N+1,l 2E N,0 ) Silvana Botti Electronic excitations & photovoltaics 6 / 34

10 Problems of standard DFT Excitation energies: energy gap Optical gap: E gap = I A E exc binding Silvana Botti Electronic excitations & photovoltaics 6 / 34

11 Problems of standard DFT A intuitive path: propagation of electrons and holes Π0 In the many-body Green s function framework: GW for electron addition and removal (one-particle G) Bethe-Salpeter equation for the inclusion of electron-hole interaction (two-particle G) The price to pay is a more involved theoretical and computational framework L. Hedin, Phys. Rev. 139 (1965) Silvana Botti Electronic excitations & photovoltaics 7 / 34

12 Problems of standard DFT Green s function and Hedin s equations Propagation of an extra particle (electron or hole): G(r 1, r 2, t 1 t 2 ) = i N T [ ˆψ(r 1, t 1 ) ˆψ (r 2, t 2 )] N Electron density: ρ (r) = G(r, r, t, t + ) Spectral function: A(ω) = 1/πTr {Im G(r 1, r 2, ω)} G has poles at addition and removal energies Silvana Botti Electronic excitations & photovoltaics 8 / 34

13 Problems of standard DFT Green s function and Hedin s equations Propagation of an extra particle (electron or hole): W W = v + vpw Σ = GWΓ P G(r 1, r 2, t 1 t 2 ) = i N T [ ˆψ(r 1, t 1 ) ˆψ (r 2, t 2 )] N Σ P = GG P = GGΓ G=G 0 +G 0 Σ G Γ G Γ=1+(δΣ/δG)GGΓ Hedin s equations should be solved self-consistently Silvana Botti Electronic excitations & photovoltaics 8 / 34

14 Problems of 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 Electronic excitations & photovoltaics 9 / 34

15 2 Cu-based delafossite transparent conductive oxides J. Vidal, S. Botti, P. Olsson, J.-F. Guillemoles, and L. Reining, Strong interplay between structure and electronic properties in CuIn(S,Se)2: a first-principles study, Phys. Rev. Lett. 104, (2010). I. Aguilera, S. Botti, J. Vidal, P. Wahnón, and L. Reining, Band structure and optical absorption of CuGaS 2 : a self-consistent GW study, to be submitted (2010). J. Vidal, Ab initio Calculations of the Electronic Properties of CuIn(S,Se) 2 and other Materials for Photovoltaic Applications, Ph.D. Thesis, Ecole Polytechnique, France (2010). Silvana Botti Electronic excitations & photovoltaics 10 / 34

16 Present state of photovoltaic efficiency Silvana Botti Electronic excitations & photovoltaics 11 / 34

17 CIGS solar cell Devices have to fulfill 2 functions: Photogeneration of electron-hole pairs Separation of charge carriers to generate a current Structure: Molybdenum back contact CIGS layer (p-type layer) CdS layer (n-type layer) ZnO:Al TCO contact Würth Elektronik GmbH & Co. Efficiency = 13 % Silvana Botti Electronic excitations & photovoltaics 12 / 34

18 Energy converted by a silicon solar cell Carriers below the gap are not absorbed Silvana Botti Electronic excitations & photovoltaics 13 / 34

19 CIGS properties Cu(In,Ga)(S,Se) 2 absorbers: high optical absorption thin-layer films optimal photovoltaic gap (record efficiency 20.1%) self-doping with native defects p-n junctions CuInS extraordinary stability under 2 12 operating conditions: tolerance to CuGaSe 2 10 large off-stoichiometries, stress, 8 defects (not well understood) E g [ev] η [%] CuInSe 2 Cu(In,Ga)Se 2 Silvana Botti Electronic excitations & photovoltaics 14 / 34

20 LDA energies 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 Electronic excitations & photovoltaics 15 / 34

21 Perturbative GW: best G, best W 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&Louie, PRB 34 (1986); Godby, Schlüter&Sham, PRB 37 (1988) Silvana Botti Electronic excitations & photovoltaics 16 / 34

22 Perturbative GW: best G, best W 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&Louie, PRB 34 (1986); Godby, Schlüter&Sham, PRB 37 (1988) Silvana Botti Electronic excitations & photovoltaics 16 / 34

23 G 0 W 0 energies 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 Electronic excitations & photovoltaics 17 / 34

24 Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al ) Hybrid functionals (e.g. Fuchs et al. 2007) E PBE0 xc = Exc PBE + 1 ( E HF x 4 Ex PBE ) Silvana Botti Electronic excitations & photovoltaics 18 / 34

25 Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al ) Hybrid functionals (e.g. Fuchs et al. 2007) E PBE0 xc = Exc PBE + 1 ( E HF x 4 Ex PBE ) Silvana Botti Electronic excitations & photovoltaics 18 / 34

26 Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al ) Hybrid functionals (e.g. Fuchs et al. 2007) Self-consistent approaches: GWscQP scheme (Faleev et al. 2004) sccohsex scheme (Hedin 1965, Bruneval et al. 2005) Silvana Botti Electronic excitations & photovoltaics 18 / 34

27 Beyond Standard GW Looking for another starting point: DFT with another approximation for v xc : GGA, EXX,... (e.g. Rinke et al. 2005) LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al ) Hybrid functionals (e.g. Fuchs et al. 2007) Self-consistent approaches: GWscQP scheme (Faleev et al. 2004) sccohsex scheme (Hedin 1965, Bruneval et al. 2005) Our choice is to get a better starting point for G 0 W 0 using sccohsex Silvana Botti Electronic excitations & photovoltaics 18 / 34

28 COHSEX: approximation to GW self-energy Statically 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) Induced classical potential due to an extra point charge: 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 )] L. Hedin and S. Lundqvist, Solid State Phys. 23, 1 (1969); Bruneval et al. PRL 97, (2006); Gatti et al. PRL 99, (2007) Silvana Botti Electronic excitations & photovoltaics 19 / 34

29 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 Electronic excitations & photovoltaics 20 / 34

30 of CuGaSe LDA sccohsex+g 0 W 0 Energy (ev) T Γ N Cu 3d - Se 4p Ga-Se bond Se 4s Comparison of LDA (red) and scgw (blue) bands of CuGaSe 2 Analogous results for CuGaS 2 No need to include the d electrons of Ga in the valence Agreement with experimental gap within less than 0.1 ev Silvana Botti Electronic excitations & photovoltaics 21 / 34

31 Stability of the gap in In compounds ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. 1. Experiments measure the same value for the gap (within 10%) 6 a [Å] LDA GGA B3LYP HSE03 HSE06 PBE0 u HF+c b u 2. The experimental dispersion of u is, however, large Only hybrid functionals give structural parameters which overlap with experiments Jaffe, PRB 29, 1882 (1984); Merino, J. Appl. Phys. 80, 5610 (1996) Jaffe, PRB 27, 5176 (1983); Jiang, Sem. Sci. Technol. 23, (2008) Silvana Botti Electronic excitations & photovoltaics 22 / 34

32 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. DFT-LDA 2 CuInS 2 E g [ev] 1 Strong variations in DFT-LDA (in agreement with literature) u Silvana Botti Electronic excitations & photovoltaics 22 / 34

33 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. DFT-LDA G 0 W 0 2 CuInS 2 E g [ev] 1 G 0 W 0 does not change the slope u Silvana Botti Electronic excitations & photovoltaics 22 / 34

34 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. DFT-LDA G 0 W 0 2 CuInS 2 E g [ev] 1... except if the gap is already open u Silvana Botti Electronic excitations & photovoltaics 22 / 34

35 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. 2 CuInS 2 DFT-LDA G 0 W 0 scgw E g [ev] 1 sc-gw enhances the gap variation u Silvana Botti Electronic excitations & photovoltaics 22 / 34

36 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. E g [ev] 2 1 CuInS 2 DFT-LDA G 0 W 0 scgw HSE06 HSE06 hybrid gives an intermediate slope E HSE06 xc = Exc GGA + 1 Ex HF,sr 4 Ex GGA,sr u Silvana Botti Electronic excitations & photovoltaics 22 / 34

37 Stability of the gap in In compounds 3 ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. E g [ev] CuInS 2 DFT-LDA G 0 W 0 scgw HSE06 HSE06 ε 0 a modified-hse06 (the mixing parameter of the screened Fock exchange is proportional to the screening) gives the sc-gw slope u Silvana Botti Electronic excitations & photovoltaics 22 / 34

38 Stability of the gap in In compounds ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. Is the gap stable under lattice distortion? sc-gw and hybrid calculations predict even stronger variations than LDA The gap is not stable under lattice distortion alone Silvana Botti Electronic excitations & photovoltaics 22 / 34

39 Stability of the gap in In compounds ( ) Anion displacement: u = 1/4 + R 2 Cu (S,Se) R2 In (S,Se) /a 2 1/4. Is the gap stable under lattice distortion? sc-gw and hybrid calculations predict even stronger variations than LDA The gap is not stable under lattice distortion alone Silvana Botti Electronic excitations & photovoltaics 22 / 34

40 Formation energy of Cu vacancies The formation energy of V Cu varies under lattice distortion: E f = E DFT f E sc GW VBM Silvana Botti Electronic excitations & photovoltaics 23 / 34

41 Formation energy of Cu vacancies The formation energy of V Cu varies under lattice distortion: E f = E DFT f E sc GW VBM conduction band minimum (CBM) valence band maximum (VBM) It is essential to go beyond DFT-LDA LDA+U (blue lines) gives only constant shifts Zhang et al. PRB 57, 9642 (1998); Lany et al. PRB 78, (2008). Silvana Botti Electronic excitations & photovoltaics 23 / 34

42 Why is the experimental gap so stable? A feedback loop can explain the stability of the band gap: u E g H f (V Cu ) [V Cu ] E g Silvana Botti Electronic excitations & photovoltaics 24 / 34

43 Why is the experimental gap so stable? A feedback loop can explain the stability of the band gap: u E g H f (V Cu ) [V Cu ] E g E g = *ln([V Cu ]/N Cu ) E g = *u E g [ev] E g [ev] VBM [ev] ln([v Cu ]/N Cu ) u E v scgw = *u u Silvana Botti Electronic excitations & photovoltaics 24 / 34

44 Why is the experimental gap so stable? A feedback loop can explain the stability of the band gap: u E g H f (V Cu ) [V Cu ] E g Experimental variation of u is 0.02 E g 0.65 ev Considering variations of u and [V Cu ] E g ev Silvana Botti Electronic excitations & photovoltaics 24 / 34

45 Cu delafossites TCOs 2 Cu-based delafossite transparent conductive oxides J. Vidal, F. Trani, F. Bruneval, M. A. L. Marques and S. Botti, Effects of polarization on the band-structure of delafossite transparent conductive oxides, Phys. Rev. Lett. 104, (2010). F. Trani, J. Vidal, S. Botti and M. A. L. Marques, of delafossite transparent conductive oxides from a self-consistent GW approach, Phys. Rev. B 82, (2010). Silvana Botti Electronic excitations & photovoltaics 25 / 34

46 Cu delafossites TCOs Delafossite TCO properties Cu(Al,In,Ga)O2 thin-films are transparent and conducting: p-type or even bipolar conductivity combination of n- and p-type TCO materials allows stacked cells with increased efficiency functional windows transparent transistors Silvana Botti Electronic excitations & photovoltaics 26 / 34

47 Cu delafossites TCOs The long dispute about delafossite gaps E g [ev] E g indirect E g direct =E g direct -Eg indirect exp. direct gap exp. indirect gap LDA LDA+U B3LYP HSE03 HSE06 G 0 W 0 scgw scgw+p Experimental data are for optical gap: exciton binding energy 0.5 ev [Laskowski et al. PRB 79, (2009)] Strong lattice polaron effects are expected 1 ev [Bechstedt et al. PRB 72, (2005)] Silvana Botti Electronic excitations & photovoltaics 27 / 34

48 Cu delafossites TCOs Comparison with hybrid functional calculations Energy(eV) sc-gw HSE03 LDA+U -2.0 Γ F L Z Γ Γ F L Z Γ Γ F L Z Γ Strong differences both in dispersion and energy gaps Are hybrids a good compromise? Silvana Botti Electronic excitations & photovoltaics 28 / 34

49 Optical spectra Bethe-Salpeter equation: electron-hole interaction The BSE for the reducible 4-point polarizability L: ( ) L = L 0 + L 4 0 v 4 W L 4 v(1, 2, 3, 4) = δ(1, 2)δ(3, 4)v(1, 3) and 4 W = δ(1, 3)δ(2, 4)W (1, 2) The measurable χ is obtained via a two-point contraction of L χ red (1, 2) = L(1, 1, 2, 2) In transition space and using the only-resonant approximation: The ingredients are: H 2p,exc (vc)(v c ) Av c λ = Eλ exc Av c λ Kohn-Sham wavefunctions and energies GW corrected energies screening matrix ε 1 GG (q) Salpeter and Bethe, Phys. Rev. 84, 1232 (1951) Silvana Botti Electronic excitations & photovoltaics 29 / 34

50 Optical spectra Optical absorption of CuGaS 2 Excitonic binding energy of about 0.05 ev Experimental optical gap at 2.5 ev 20 Exp. 1 Optical absorption scgw gap Exp. 2 BSE calculations Energy (ev) Silvana Botti Electronic excitations & photovoltaics 30 / 34

51 Optical spectra Optical absorption of CuInO 2 Strong excitonic effects also for the In compound Exp. absorption edge at 3.9 ev Polaronic effects should also be strong RPA (NLF) scgw(nlf) solid lines: xy component, dashed: z component ε scgw+bse Energy (ev) Silvana Botti Electronic excitations & photovoltaics 31 / 34

52 Conclusions and perspectives Conclusions and perspectives Interpretation of experiments is often not straightforward many coupled effects! Methods that go beyond ground-state DFT are by now well established A better starting point than LDA is absolutely necessary for d-electrons Self-consistent COHSEX+G 0 W 0 gives a very good description of quasi-particle states Hybrid functionals can be a good compromise LDA+U does not work when there is hybridization of p d states In progress now: Absorption spectra from the Bethe-Salpeter equation for all compounds Defects using VASP and improved hybrid functionals Silvana Botti Electronic excitations & photovoltaics 32 / 34

53 Conclusions and perspectives Building-Integrated Photovoltaics Projects realised so far do not yet represent the full range of products available on the market: these include integratable crystalline modules, thin-layer modules, transparent and shading modules, solar roof tiles, photovoltaic roof foils or complete solar roofs Playing with light and glass: building-integrated photovoltaics opens new creative opportunities for architects Silvana Botti Electronic excitations & photovoltaics 33 / 34

54 Thanks! Thanks to all collaborators! Thank you! LSI École Polytechnique LPMCN Université Lyon 1 Lucia Reining Miguel Marques Julien Vidal Fabio Trani Irene Aguilera Guilherme Vilhena David Kammerlander IRDEP Paris J.-F. Guillemoles Pär Olsson CEA Saclay Fabien Bruneval Silvana Botti Electronic excitations & photovoltaics 34 / 34