Theoretical spectroscopy

Transcription

1 Theoretical spectroscopy from basic developments to real-world applications M. A. L. Marques 1 LPMCN, CNRS-Université Lyon 1, France 2 European Theoretical Spectroscopy Facility M. A. L. Marques 1 / 30

2 Outline 1 Recent advances in theoretical spectroscopy 2 A real world example: delafossite transparent conductive oxides 3 Can we make it faster? M. A. L. Marques 2 / 30

3 Recent advances in theoretical spectroscopy 1 Recent advances in theoretical spectroscopy M.A.L. Marques and A. Rubio (guest editors), Time-dependent density-functional theory, Phys. Chem. Chem. Phys. 11, (2009). M.A.L. Marques, C. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross (Eds.), Time-dependent density-functional theory, Lecture Notes in Physics, Vol. 706 (Springer, Berlin, 2006). G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, (2002). M. A. L. Marques 3 / 30

4 Recent advances in theoretical spectroscopy Density-functional theory The current work-horse theory to study atoms, molecules, and solids is density-functional theory (DFT). This is clearly one of the most successful theories in Physics of the last 50 years. ] ɛψ = [ v ext + v Hartree + v xc ψ For spectroscopy, one often has to resort to its time-dependent extension, time-dependent DFT. i [ ] t ψ = v ext + v Hartree + v xc ψ M. A. L. Marques 4 / 30

5 Recent advances in theoretical spectroscopy Density-functional theory The current work-horse theory to study atoms, molecules, and solids is density-functional theory (DFT). This is clearly one of the most successful theories in Physics of the last 50 years. ] ɛψ = [ v ext + v Hartree + v xc ψ For spectroscopy, one often has to resort to its time-dependent extension, time-dependent DFT. i [ ] t ψ = v ext + v Hartree + v xc ψ M. A. L. Marques 4 / 30

6 Recent advances in theoretical spectroscopy Example I: superconductivity DFT often yields excellent phonons and electron-phonon couplings, and can therefore be used to study BCS superconductivity. An example: BaSi 2 J. Flores-Livas, R. Debord, S. Botti, A. San Miguel, MALM, and S. Paillès, submitted (2010) M. A. L. Marques 5 / 30

7 Recent advances in theoretical spectroscopy Example I: superconductivity DFT often yields excellent phonons and electron-phonon couplings, and can therefore be used to study BCS superconductivity. An example: BaSi 2 J. Flores-Livas, R. Debord, S. Botti, A. San Miguel, MALM, and S. Paillès, submitted (2010) M. A. L. Marques 5 / 30

8 Recent advances in theoretical spectroscopy Example II: atomic clusters TDDFT is now routinely used to study absorption spectra of atomic clusters. An example: relativistic effects in gold clusters A. Castro, MALM, A.H. Romero, M.J.T. Oliveira, and A. Rubio J. Chem. Phys. 129, (2008) M. A. L. Marques 6 / 30

9 Recent advances in theoretical spectroscopy Example II: atomic clusters TDDFT is now routinely used to study absorption spectra of atomic clusters. An example: relativistic effects in gold clusters 6 8 n=2 n= n=4 n= n= E (ev) n= E (ev) E (ev) E (ev) A. Castro, MALM, A.H. Romero, M.J.T. Oliveira, and A. Rubio J. Chem. Phys. 129, (2008) M. A. L. Marques 6 / 30

10 Recent advances in theoretical spectroscopy Example III: Biochemistry Furthermore, by now TDDFT is the only ab-initio tool that is able to tackle large biological systems. An example: bioluminescence of fireflies B.F. Milne, MALM, and F. Nogueira, Phys. Chem. Chem. Phys. 12, (2010) D. Cai, MALM, B.F. Milne, and F. Nogueira, J. Phys. Chem. Lett. 1, (2010) M. A. L. Marques 7 / 30

11 Recent advances in theoretical spectroscopy Example III: Biochemistry Furthermore, by now TDDFT is the only ab-initio tool that is able to tackle large biological systems. An example: bioluminescence of fireflies B.F. Milne, MALM, and F. Nogueira, Phys. Chem. Chem. Phys. 12, (2010) D. Cai, MALM, B.F. Milne, and F. Nogueira, J. Phys. Chem. Lett. 1, (2010) M. A. L. Marques 7 / 30

12 Recent advances in theoretical spectroscopy Example III: Biochemistry Furthermore, by now TDDFT is the only ab-initio tool that is able to tackle large biological systems. An example: bioluminescence of fireflies B.F. Milne, MALM, and F. Nogueira, Phys. Chem. Chem. Phys. 12, (2010) D. Cai, MALM, B.F. Milne, and F. Nogueira, J. Phys. Chem. Lett. 1, (2010) M. A. L. Marques 7 / 30

13 Recent advances in theoretical spectroscopy Solids: the Kohn-Sham band gap underestimation Standard DFT yields excellent structural properties but energy gaps are, at best half of experiment Hybrid functionals solve to a large extent this problem but their accuracy is very system-dependent van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 8 / 30

14 Recent advances in theoretical spectroscopy Solids: the Kohn-Sham band gap underestimation Standard DFT yields excellent structural properties but energy gaps are, at best half of experiment Hybrid functionals solve to a large extent this problem but their accuracy is very system-dependent van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 8 / 30

15 Recent advances in theoretical spectroscopy Solids: the Kohn-Sham band gap underestimation Standard DFT yields excellent structural properties but energy gaps are, at best half of experiment Hybrid functionals solve to a large extent this problem but their accuracy is very system-dependent van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 8 / 30

16 Recent advances in theoretical spectroscopy Solids: the Kohn-Sham band gap underestimation Standard DFT yields excellent structural properties but energy gaps are, at best half of experiment Hybrid functionals solve to a large extent this problem but their accuracy is very system-dependent van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 8 / 30

17 Recent advances in theoretical spectroscopy Standard solution: Hedin s GW The GW equations reformulate many-body perturbation theory in terms of the screened Coulomb interaction W. W W = v + vpw Σ = GWΓ P Σ P = GGΓ G=G 0 +G 0 Σ G Γ G Γ=1+(δΣ/δG)GGΓ M. A. L. Marques 9 / 30

18 Recent advances in theoretical spectroscopy Standard solution: Hedin s GW The GW equations reformulate many-body perturbation theory in terms of the screened Coulomb interaction W. W Σ = GWΓ Σ G=G 0 +G 0 Σ G G W = v + vpw P P = GG P = GGΓ Γ Γ=1+(δΣ/δG)GGΓ M. A. L. Marques 9 / 30

19 Recent advances in theoretical spectroscopy Perturbative GW In many materials, perturbative GW correct the LDA band gaps, and gives reasonable agreement with experimental data. In most metal oxides, however, quasiparticle wavefunctions are badly reproduced by the LDA, so the GW correction is too small. van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 10 / 30

20 Recent advances in theoretical spectroscopy Perturbative GW In many materials, perturbative GW correct the LDA band gaps, and gives reasonable agreement with experimental data. In most metal oxides, however, quasiparticle wavefunctions are badly reproduced by the LDA, so the GW correction is too small. van Schilfgaarde, Kotani, and Faleev, Phys. Rev. Lett. 96, (2006) M. A. L. Marques 10 / 30

21 Recent advances in theoretical spectroscopy Real solution: self-consistent GW A recipe to perform self-consistent GW has been given by Faleev, Kotani, and van Schilfgaarde. It is based on a procedure to make the self-energy static and Hermitian. We normally prefer a similar approach, that turns out to yield very similar results, but is lighter from the computational point of view: start from an LDA band-structure and perform a self-consistent COHSEX (Coulomb Hole and Screened Exchange) calculation. add dynamical effects by performing a perturbative GW step on top of the self-consistent COHSEX. This approach gives excellent results for many complex compounds, such as CuIn(S,Se) 2, Cu 2 O, VO 2, etc. S.V. Faleev, Phys. Rev. Lett. 93, (2004) F. Bruneval et al. Phys. Rev. Lett. 97, (2006) M. A. L. Marques 11 / 30

22 A real world example: delafossite transparent conductive oxides 2 A real world example: 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, Band structures of delafossite transparent conductive oxides from a self-consistent GW approach, Phys. Rev. B 82, (2010). M. A. L. Marques 12 / 30

23 A real world example: delafossite transparent conductive oxides The delafossite crystal structure Edge-sharing MO 6 octahedral layers CuO 2 dumbbell layers M is a group-iii element (Al, Ga, In, B) M. A. L. Marques 13 / 30

24 A real world example: delafossite transparent conductive oxides 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 very similar structure of the Cu(Ga,In)(S,Se)2 family of absorbers for thin-film photovoltaics. M. A. L. Marques 14 / 30

25 A real world example: delafossite transparent conductive oxides Literature - early results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P-type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Appl. Phys. Lett. 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p-n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films: low-energy indirect gap, direct gap comparable to other TCOs. Gap (ev) Indirect Direct CuAlO CuInO CuGaO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band-gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M M. A. L. Marques 15 / 30

26 A real world example: delafossite transparent conductive oxides Literature - early results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P-type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Appl. Phys. Lett. 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p-n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films: low-energy indirect gap, direct gap comparable to other TCOs. Gap (ev) Indirect Direct CuAlO CuInO CuGaO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band-gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M M. A. L. Marques 15 / 30

27 A real world example: delafossite transparent conductive oxides Literature - early results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P-type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Appl. Phys. Lett. 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p-n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films: low-energy indirect gap, direct gap comparable to other TCOs. Gap (ev) Indirect Direct CuAlO CuInO CuGaO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band-gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M M. A. L. Marques 15 / 30

28 A real world example: delafossite transparent conductive oxides Literature - early results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P-type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Appl. Phys. Lett. 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p-n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films: low-energy indirect gap, direct gap comparable to other TCOs. Gap (ev) Indirect Direct CuAlO CuInO CuGaO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band-gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M M. A. L. Marques 15 / 30

29 A real world example: delafossite transparent conductive oxides Literature - early results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P-type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Appl. Phys. Lett. 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p-n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films: low-energy indirect gap, direct gap comparable to other TCOs. Gap (ev) Indirect Direct CuAlO CuInO CuGaO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band-gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M M. A. L. Marques 15 / 30

30 A real world example: delafossite transparent conductive oxides Literature - recent results 2001 Yanagi et al. [J. of Appl. Phys. 88, 4159 (2001)] Photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations there is no trace of low-energy indirect bandgaps 2006 Pellicer-Porres et al. [Appl. Phys. Lett. 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Laskowski et al. [Phys. Rev. B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5 ev Pellicer-Porres et al. [Semicond. Sci. Technol. 24, (2009)] Large polaronic constant in CuAlO 2. M. A. L. Marques 16 / 30

31 A real world example: delafossite transparent conductive oxides Literature - recent results 2001 Yanagi et al. [J. of Appl. Phys. 88, 4159 (2001)] Photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations there is no trace of low-energy indirect bandgaps 2006 Pellicer-Porres et al. [Appl. Phys. Lett. 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Laskowski et al. [Phys. Rev. B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5 ev Pellicer-Porres et al. [Semicond. Sci. Technol. 24, (2009)] Large polaronic constant in CuAlO 2. M. A. L. Marques 16 / 30

32 A real world example: delafossite transparent conductive oxides Literature - recent results 2001 Yanagi et al. [J. of Appl. Phys. 88, 4159 (2001)] Photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations there is no trace of low-energy indirect bandgaps 2006 Pellicer-Porres et al. [Appl. Phys. Lett. 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Laskowski et al. [Phys. Rev. B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5 ev Pellicer-Porres et al. [Semicond. Sci. Technol. 24, (2009)] Large polaronic constant in CuAlO 2. M. A. L. Marques 16 / 30

33 A real world example: delafossite transparent conductive oxides Literature - recent results 2001 Yanagi et al. [J. of Appl. Phys. 88, 4159 (2001)] Photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations there is no trace of low-energy indirect bandgaps 2006 Pellicer-Porres et al. [Appl. Phys. Lett. 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Laskowski et al. [Phys. Rev. B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5 ev Pellicer-Porres et al. [Semicond. Sci. Technol. 24, (2009)] Large polaronic constant in CuAlO 2. M. A. L. Marques 16 / 30

34 A real world example: delafossite transparent conductive oxides Literature - recent results 2001 Yanagi et al. [J. of Appl. Phys. 88, 4159 (2001)] Photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations there is no trace of low-energy indirect bandgaps 2006 Pellicer-Porres et al. [Appl. Phys. Lett. 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Laskowski et al. [Phys. Rev. B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5 ev Pellicer-Porres et al. [Semicond. Sci. Technol. 24, (2009)] Large polaronic constant in CuAlO 2. M. A. L. Marques 16 / 30

35 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band-structure I Upon going from LDA to G 0 W 0 to scgw, there is an upshift of the conduction band at Γ with respect to L. Within scgw, the minimum of the conduction band is at L, the direct and indirect band gaps are almost identical Our results were confirmed by a recent, independent calculation reported by Christensen et al. Phys. Rev. B 81, (2010) The scgw direct gap is more than 1 ev higher than experiment! M. A. L. Marques 17 / 30

36 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band-structure II scgw corrections are strongly k-dependent. No scissor operator possible. 8.0 Hybrids (HSE03) show a large difference with scgw in the conduction band, while the valence bands are very accurate. LDA+U describes incorrectly both the gap and the band dispersion. Energy(eV) sc-gw HSE03 LDA+U Γ F L Z Γ Γ F L Z Γ Γ F L Z Γ M. A. L. Marques 18 / 30

37 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band-structure II According to the experimental data, the polaronic constant in CuAlO 2 is 1. In delafossites we have non negligible contribution of the lattice polarization to the electronic screening. Lattice polarization effects can be included within GW, simply adding the ionic contribution to the dielectric matrix in the calculation of the screened potential W. ɛ GG (q, ω) = δ GG + 4πP GG (q, ω) + 4πP lat GG(q, ω) Using the long wavelength contribution of the lattice polarizability P lat 00(q 0, ω), and the Lyddane-Sachs-Teller relationship for the static case, a simple model for the lattice polarizability is derived from static dielectric constant ɛ 0 low-frequency dielectric constant ɛ zone center optical frequencies ω LO and ω TO Bechstedt et al. Phys. Rev. B 72, (2005) M. A. L. Marques 19 / 30

38 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band gaps Experimental data are for optical gap: exciton binding energy 0.5 ev The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher no trace of an small indirect gap Agreement with experiment can only be achived by the addition of lattice polarization effects. 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 M. A. L. Marques 20 / 30

39 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band gaps Experimental data are for optical gap: exciton binding energy 0.5 ev The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher no trace of an small indirect gap Agreement with experiment can only be achived by the addition of lattice polarization effects. 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 M. A. L. Marques 20 / 30

40 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band gaps Experimental data are for optical gap: exciton binding energy 0.5 ev The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher no trace of an small indirect gap Agreement with experiment can only be achived by the addition of lattice polarization effects. 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 M. A. L. Marques 20 / 30

41 A real world example: delafossite transparent conductive oxides Results: CuAlO 2 band gaps Experimental data are for optical gap: exciton binding energy 0.5 ev The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher no trace of an small indirect gap Agreement with experiment can only be achived by the addition of lattice polarization effects. 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 M. A. L. Marques 20 / 30

42 A real world example: delafossite transparent conductive oxides Results: CuMO 2 band gaps E GW ind E GW dir E GW dir -E GW ind E GW dir -E LDA dir CuBO (a) CuBO (b) CuAlO CuGaO CuInO (a) LDA-optimized geometry; (b) Experimental geometry The direct to indirect band gap difference increases on increasing the M atomic number. Apart from the CuBO 2, scgw adds to the LDA direct band gap a constant amount of about 2.5 ev. M. A. L. Marques 21 / 30

43 Can we make it faster? 3 Can we make it faster? M. A. L. Marques, J. Vidal, M. J. T. Oliveira, L. Reining, and S. Botti, On the mixing parameter of hybrid functionals, submitted (2010). M. J. T. Oliveira, E. Räsänen, S. Pittalis, M. A. L. Marques Toward an all-round semi-local potential for the electronic exchange, accepted (2010). M. A. L. Marques 22 / 30

44 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

45 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

46 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

47 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

48 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

49 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

50 Can we make it faster? In the field that is now called Materials Design, one typically screens materials for specific properties. To perform such calculations one needs a method that yields good spectroscopy properties (like band-gaps or absorption edges) at a modest computational effort. Self-consistent GW is too expensive. Can we simplify it? We are faced with two problems: Exchange The calculation of the Coulomb integrals is very time-consuming New meta-ggas based on the Becke-Johnson functional (Tran-Blaha, Rasanen-Pittalis-Proeto, etc.) may be a solution Correlation Screening! Hybrid functionals seem to provide a good compromise between accuracy and computational effort. M. A. L. Marques 23 / 30

51 Can we make it faster? What is the hybrid mixing parameter? Let us look at the quasi-particle equation: ] [ v ext(r) + v H (r) φ QP i (r) + d 3 r Σ(r, r ; ε QP i And now let us look at the different approximations: COHSEX: Hybrids occ Σ = occ Σ = i i φ QP i φ QP i So, we infer that a 1/ɛ! )φ QP i (r ) = ε QP i φ QP (r)φ QP (r )W (r, r ; ω = 0) + δ(r r )Σ COH (r) i (r)φ QP (r )a v(r r ) + δ(r r )(1 a) v DFT (r) i i (r ) M. A. L. Marques 24 / 30

52 Can we make it faster? Does it work (a = 1/ɛ )? Theoretical gap (ev) Si, MoS GaAs 2 SiC,CdS,AlP GaN ZnS ZnO Ge y=x PBE PBE0 PBE0ε C BN, AlN MgO Xe LiCl SiO Kr Ar, LiF Experimental gap (ev) Ne Errors: PBE: 47% Hartree-Fock: 250% PBE0: 29% PBE0ɛ : 16.53% M. A. L. Marques 25 / 30

53 Can we make it faster? Can we do better? Screening is related to the gap. So, if we have an estimator of the gap of the material, we can also get an estimator of the dielectric constant. There are several local estimators on the market: G = 1 8 n 2 /n 2 (Gutle et al. 1999) n /n (Heyd et al. 2003; Krukau et al. 2008) τ W = n 2 /8n (Jaramillo et al. 2003) These are however, local estimators and we need a global estimator. The solution is averaging. We follow the idea of the Tran and Blaha meta-gga and define ḡ = 1 d 3 n(r) r V cell cell n(r) We then fit ḡ F Tran and P Blaha, Phys. Rev. Lett. 102, (2009) M. A. L. Marques 26 / 30

54 Can we make it faster? Hybrids results Theoretical gap (ev) Si, MoS GaAs 2 SiC,CdS,AlP GaN ZnS ZnO Ge y=x PBE PBE0 PBE0ε PBE0 mix TB09 C BN, AlN MgO Xe LiCl SiO Kr Ar, LiF Experimental gap (ev) Ne Errors: PBE0: 29% PBE0ɛ : 16.53% PBE0 mix : 14.37% TB09: 9.85 G 0 W 0 : M. A. L. Marques 27 / 30

55 Can we make it faster? How about HSE? Theoretical gap (ev) Si, MoS GaAs 2 SiC,CdS,AlP GaN ZnS ZnO Ge y=x HSE06 HSE06 mix C BN, AlN MgO Xe LiCl SiO Kr Ar, LiF Experimental gap (ev) Ne Errors: PBE0: 29% PBE0 mix : 14.37% HSE06: 16.92% HSE06 mix : 10.36% TB09: 9.85 G 0 W 0 : M. A. L. Marques 28 / 30

56 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

57 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

58 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

59 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

60 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

61 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

62 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

63 Conclusions Can we make it faster? Theoretical spectroscopy has come a long way: Many different spectroscopic quantities can be obtained for a variety of systems for finite systems (molecules, clusters) even the simple LDA can often lead to very satisfactory spectra (absorption, dychroism, non-linear polarizabilities, etc.) for solids, standard DFT yields excellent phonons, electron-phonon couplings, etc. Band structures traditionally are accessed by GW methods, but for transition metal oxydes we need to do better (self-consistent GW, e.g.) and we should not forget the phonons! Hybrids, or maybe the new meta-ggas seem to be a very good compromise Several very new recent developments in this area in the last couple of years! M. A. L. Marques 29 / 30

64 Thanks! Thanks to all collaborators! Thank you! LSI École Polytechnique Lucia Reining Julien Vidal LPMCN Université Lyon 1 Silvana Botti Fabio Trani Guilherme Vilhena David Kammerlander CEA Saclay Fabien Bruneval M. A. L. Marques 30 / 30