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Electronic Properties of Materials for Solar Cells: Which Ab Initio Approaches Can We Trust?

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Electronic Properties of Materials for Solar Cells: Which Ab Initio Approaches Can We Trust? Electronic properties of materials for solar cells: Which ab initio approaches can we trust? Silvana Botti 1LSI, CNRS-CEA-Ecole´ Polytechnique, Palaiseau, France 2LPMCN, CNRS-Universite´ Lyon 1, France 3European Theoretical Spectroscopy Facility September 14, 2009 – CONARES, Helsinki Silvana Botti Electronic excitations in solar cells 1 / 57 Collaborators Ecole Polytechnique Universite´ Lyon 1 Julien Vidal, Lucia Reining Fabio Trani, Miguel Marques EDF Paris CEA Saclay Par¨ Olsson, J.-F. Guillemoles Fabien Bruneval Silvana Botti Electronic excitations in solar cells 2 / 57 European Theoretical Spectroscopy Facility (ETSF) The ETSF is a knowledge center for theoretical spectroscopy carrying out state-of-the-art research on theoretical and computational methods for studying electronic and optical properties of materials. The ETSF gathers the experience and know-how of more than 200 researchers in Europe and the United States. The ETSF offers its expertise to researchers, industry, and students in the form of collaborative projects, free scientific software and training. Proposals can be submitted at any moment! Further information: www.etsf.eu Silvana Botti Electronic excitations in solar cells 3 / 57 Outline 1 Thin-film photovoltaic materials 2 What can we calculate within standard DFT? 3 How to go beyond standard DFT? GW approaches! 4 How to compare with experiments? Silvana Botti Electronic excitations in solar cells 4 / 57 Thin-film photovoltaic materials Outline 1 Thin-film photovoltaic materials 2 What can we calculate within standard DFT? 3 How to go beyond standard DFT? GW approaches! 4 How to compare with experiments? Silvana Botti Electronic excitations in solar cells 5 / 57 Thin-film photovoltaic materials Present state of photovoltaic efficiency from National Renewable Energy Laboratory (USA) Silvana Botti Electronic excitations in solar cells 6 / 57 Thin-film photovoltaic materials 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) Wurth¨ Elektronik GmbH & Co. ZnO:Al TCO contact Efficiency = 13 % Silvana Botti Electronic excitations in solar cells 7 / 57 Thin-film 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 Electronic excitations in solar cells 8 / 57 What can we calculate within standard DFT? Outline 1 Thin-film photovoltaic materials 2 What can we calculate within standard DFT? 3 How to go beyond standard DFT? GW approaches! 4 How to compare with experiments? Silvana Botti Electronic excitations in solar cells 9 / 57 What can we calculate within standard DFT? Ground state densities vs potentials At the heart of density functional theory (DFT) Is there a 1-to-1 mapping between different external potentials v(r) and their corresponding ground state densities ρ(r)? Silvana Botti Electronic excitations in solar cells 10 / 57 What can we calculate within standard DFT? Density functional theory (DFT) If we can give a positive answer, then it can be proved that (i) all observable quantities of a quantum system are completely determined by the density. (ii) which means that the basic variable is no more the many-body wavefunction Ψ(fr)g but the electron density ρ(r). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). You can find details in R. M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer (Berlin, 1990). Silvana Botti Electronic excitations in solar cells 11 / 57 What can we calculate within standard DFT? Density functional theory (DFT) If we can give a positive answer, then it can be proved that (i) all observable quantities of a quantum system are completely determined by the density. (ii) which means that the basic variable is no more the many-body wavefunction Ψ(fr)g but the electron density ρ(r). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). You can find details in R. M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer (Berlin, 1990). Silvana Botti Electronic excitations in solar cells 11 / 57 What can we calculate within standard DFT? Density functional theory DFT in its standard form is a ground state theory: Structural parameters: lattice parameters, internal distortions are usually good in LDA or GGA Formation energies for defects calculated from total energies are often reliable ::: but ::: Kohn-Sham energies are not meant to reproduce quasiparticle band structures Kohn-Sham DOS is not meant to reproduce photoemission Silvana Botti Electronic excitations in solar cells 12 / 57 What can we calculate within standard DFT? Kohn-Sham band structure Kohn-Sham (KS) equations r2 − + v (r) 'KS (r) = "KS'KS (r) 2 KS i i i occ. X KS 2 ρ (r) = j'i (r) j i The KS states are not one-electron energy states for the quasi-electrons in the solid However, it is common to interpret the solutions of the Kohn-Sham equations as one-electron states Often one obtains good band dispersions but band gaps are systematically underestimated Silvana Botti Electronic excitations in solar cells 13 / 57 What can we calculate within standard DFT? Discontinuity in Vxc .6 .6 E = (E − E ) − (E − E ) ε ε g (N+1) (N) (N) (N−1) ε .6 1 1 KS KS ∆ = " (N + 1) − " (N) ;& N+1 N .6 ( ε JDS 1 .6 ( ε .6 JDS 1 DFT KS KS Eg = "N+1(N) − "N (N) N N 1HOHFWURQV 1HOHFWURQV DFT (N+1) (N) ∆xc = Eg − Eg = Vxc (r) − Vxc (r) Band gap error not due to LDA, but to the discontinuity in the exact Vxc L. J. Sham and M. Schluter,¨ PRL 51, 1888 (1983); PRB 32, 3883 (1985) J. P. Perdew and M. Levy, PRL 51, 1884 (1983) R. W. Godby, M. Schluter¨ and L. J. Sham, PRL 56, 2415 (1986) Silvana Botti Electronic excitations in solar cells 14 / 57 What can we calculate within standard DFT? An intuitive Picture for Absorption c unoccupied states occupied states v KS KS Independent particle KS picture using "i and 'i Silvana Botti Electronic excitations in solar cells 15 / 57 What can we calculate within standard DFT? An intuitive Picture for Absorption c unoccupied states occupied states v Photoemission process: hν − (Ekin + φ) = EN−1;v − EN;0 = −"v Silvana Botti Electronic excitations in solar cells 16 / 57 What can we calculate within standard DFT? An intuitive Picture for Absorption c unoccupied states occupied states v Optical absorption: electron-hole interaction (excitons) Silvana Botti Electronic excitations in solar cells 17 / 57 What can we calculate within standard DFT? Software supporting DFT Abinit EXCITING OpenMX ADF Fireball ORCA AIMPRO FSatom - list of codes ParaGauss Atomistix Toolkit GAMESS (UK) PLATO CADPAC GAMESS (US) PWscf (Quantum- CASTEP GAUSSIAN ESPRESSO) CPMD JAGUAR Q-Chem CRYSTAL06 MOLCAS SIESTA DACAPO MOLPRO Spartan DALTON MPQC S/PHI/nX deMon2K NRLMOL TURBOMOLE DFT++ NWChem VASP DMol3 OCTOPUS WIEN2k http://en.wikipedia.org/wiki/Density_functional_theory Silvana Botti Electronic excitations in solar cells 18 / 57 What can we calculate within standard DFT? The code ABINIT http://www.abinit.org ”First-principles computation of material properties : the ABINIT software project.” X. Gonze et al, Computational Materials Science 25, 478-492 (2002). ”A brief introduction to the ABINIT software package.” X. Gonze et al, Zeit. Kristallogr. 220, 558-562 (2005). Silvana Botti Electronic excitations in solar cells 19 / 57 What can we calculate within standard DFT? LDA Kohn-Sham energy gaps 8 AlN SrO diamond 6 ZnO,GaN,ZnS 4 ZnSe,CuBr AlAs,GaP,SiC,AlP,CdS Se,Cu2O MgO InP,GaAs,CdTe,AlSb 2 Si InN,Ge,GaSb,CdO CaO calculated gap (eV) InSb,P,InAs HgTe 0 :LDA experimental gap (eV) van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006) Silvana Botti Electronic excitations in solar cells 20 / 57 What can we calculate within standard DFT? LDA Kohn-Sham energy gaps for CIS CuInS2 DFT-LDA exp. Eg -0.11 1.54 In-S 6.5 6.9 S s band 12.4 12.0 In 4 d band 14.6 18.2 CuInSe2 DFT-LDA exp. Eg -0.29 1.05 In-Se 5.8 6.5 Se s band 12.6 13.0 In 4 d band 14.7 18.0 www.abinit.org Silvana Botti Electronic excitations in solar cells 21 / 57 What can we calculate within standard DFT? Why do we need to go 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 Electronic excitations in solar cells 22 / 57 How to go beyond standard DFT? GW approaches! Outline 1 Thin-film photovoltaic materials 2 What can we calculate within standard DFT? 3 How to go beyond standard DFT? GW approaches! 4 How to compare with experiments? Silvana Botti Electronic excitations in solar cells 23 / 57 24 / 57 for the inclusion of ¡ ¡ ¢ £ ¢ £ £ ¡ ¡ ¡ ¡ ¢ £ ¢ ¢ ¢ ¡ £ £ £ £ ¢ ¡ ¡ ¡ ¡ ¢ ¢¤£ ¢ ¡ ¢ Electronic excitations in solar cells for quasi-particle properties ¢ ¢ ¢ ¡ ¡¢¡ ¡¢¡ ¡ ¢ GW Bethe-Salpeter equation electron-hole interaction ¢ ¢ ¢ ¢ ¡ ¢ ¢ ¢ ¡¥¡ ¡ ¨ Π In the many-body framework, wethese know problems: how to solve The first step can bethan substantially the more second, complicated so inGW the following we will focus on ¢ 0 ¢ ¢ ¨ ¡¢¡ ¡ ¡ ¢ ¢ ¨ ¨ ¢ ¢ ¢ ¢ ¡ Silvana Botti How to go beyond standard DFT? GW approaches! ¡ ¢ ¡ ¢ Solution How to go beyond standard DFT? GW approaches! Hedin’s equations Σ G=G Γ 0 GW +G 0 Σ Σ = G W G Γ W = v + vPW G)GG Γ Γ=1+(δΣ/δ P P = GGΓ L. Hedin, Phys. Rev. 139 (1965). Silvana Botti Electronic
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