Optical properties of SnO 2

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1 Optial properties of SnO 2 Arjan Berger Laboratoire des Solides Irradiés Eole Polytehnique, Palaiseau, Frane European Theoretial Spetrosopy Faility (ETSF) Moving theory to appliations, 22-10, Palaiseau

2 Outline SnO 2 fats Motivation Results Conlusions

9 ev Color : Transparent (visible: 1.5-3.

3 SnO 2 Crystal Struture: Rutile Band gap: 3.6 ev Optial gap: 3.9 ev Color : Transparent (visible: ev) Doped with Sb or F ions SnO 2 is a transparent onduting oxides (TCO): eletroluminesent devies

4 Motivation SnO 2 is a prototype material for the development of TCO s. In reality not always perfetly transparent: Study optial properties of SnO 2 = reliable quasipartile band struture.

5 Motivation GW: powerful tool to alulate quasi-partile energies of solids Σ = igw W = ɛ 1 RPA v ɛ RPA = 1 v χ 0 M. van Shilfgaarde, T. Kotani, and S. Faleev, PRL 96, (2006). - GW is omputationally demanding. - Large number of states in SOS expressions for χ 0 and Σ.

6 The Polarizability Standard alulation of χ 0 and Σ: sum over states (SOS) o χ 0 GG (q, ω) = v v e i(q+g) r e i(q+g ) r v ω (ɛ ɛ v ) + iη + A.R.

7 The Polarizability Standard alulation of χ 0 and Σ: sum over states (SOS) o χ 0 GG (q, ω) = v v e i(q+g) r e i(q+g ) r v ω (ɛ ɛ v ) + iη + A.R. Advantages: Systemati and ontrollable Easy to implement Disadvantages: Huge summation over states Slow (saling=n N v N 2 G )

8 The Polarizability Standard alulation of χ 0 and Σ: sum over states (SOS) o χ 0 GG (q, ω) = v v e i(q+g) r e i(q+g ) r v ω (ɛ ɛ v ) + iη + A.R. Advantages: Systemati and ontrollable Easy to implement Disadvantages: Huge summation over states Slow (saling=n N v N 2 G ) Aim: - Keep advantages of SOS - Get rid of disadvantages: oupied states only

9 Four Criteria for GW Systemati and ontrollable Easy to implement No summation over states Better saling properties than SOS

10 Four Criteria for GW Systemati and ontrollable Easy to implement No summation over states Better saling properties than SOS Other GW approahes with less or no states: COHSEX: Extrapolar method: Sternheimer equation: L. Hedin, PR A796 (1965) F. Bruneval and X. Gonze, PRB 78, (2008). L. Reining et al., PRB 56, R4302 (1997) P. Umari et al., PRB (2010) F. Giustino et al., PRB (2010)

11 Four Criteria for GW Systemati and ontrollable Easy to implement No summation over states Better saling properties than SOS Other GW approahes with less or no states: COHSEX: Extrapolar method: Sternheimer equation: L. Hedin, PR A796 (1965) F. Bruneval and X. Gonze, PRB 78, (2008). L. Reining et al., PRB 56, R4302 (1997) P. Umari et al., PRB (2010) F. Giustino et al., PRB (2010) So far: no GW approah whih satisfies all riteria simultaneously

12 Σ GW with sum over oupied states only The bottlenek in the alulation of Σ GW : n Σ (ω) n = q,g,g W j GG (q) j n e i(q+g) r e i(q+g ) r n ω ω j ɛ

13 Σ GW with sum over oupied states only The bottlenek in the alulation of Σ GW : n Σ (ω) n = q,g,g W j GG (q) j n e i(q+g) r e i(q+g ) r n ω ω j ɛ There exists an effetive energy δ nj (q, G, G, ω) suh that n e i(q+g) r e i(q+g ) r n ω ω j ɛ Closure relation: ( n ei(q+g) r ) e i(q+g ) r n = ω ω j δ nj (q, G, G, ω) = 1 o v v v

14 Σ GW with sum over oupied states only The bottlenek in the alulation of Σ GW : n Σ (ω) n = q,g,g W j GG (q) j n e i(q+g) r e i(q+g ) r n ω ω j ɛ There exists an effetive energy δ nj (q, G, G, ω) suh that n e i(q+g) r e i(q+g ) r n ω ω j ɛ Closure relation: ( n ei(q+g) r ) e i(q+g ) r n = ω ω j δ nj (q, G, G, ω) = 1 o v v v = EET: Effetive Energy Tehnique

15 A hierarhy of approximations for δ nj An iterative sheme leads to simple approximations for δ nj (ω) (G = G ): δ n (0) q + G 2 = ɛ n + 2 δ n (1) q + G 2 = ɛ n + 2 δ (2) q + G 2 nj (ω) = ɛ n f n ρj fn ρρ + f n ρj ω ω j fn ρρ ω ω j [ɛ n + q+g 2 [ ɛ n + q+g f ρj n fn ρρ 2 + f n jj fn ρj ] ]

16 A hierarhy of approximations for δ nj An iterative sheme leads to simple approximations for δ nj (ω) (G = G ): δ n (0) q + G 2 = ɛ n + 2 δ n (1) q + G 2 = ɛ n + 2 δ (2) q + G 2 nj (ω) = ɛ n f n ρj fn ρρ + f n ρj ω ω j fn ρρ ω ω j [ɛ n + q+g 2 [ ɛ n + q+g f ρj n fn ρρ 2 + f n jj fn ρj -The fn ρρ, fn ρj, fn jj, are simple with sums over oupied states only. [ o ] fn ρj = n i n n e i(q+g) r v v e i(q+g) r [i ] n (q + G) v -From δ n (1) onwards exat for the homogeneous eletron gas. -δ (2) nj (ω) simple but nontrivial due to frequeny dependene. ] ]

17 Getting δ Starting from the definition: S n e i(q+g) r e i(q+g) r n ω ω j ɛ = n e i(q+g) r e i(q+g) r n ω ω j δ

18 Getting δ Starting from the definition: S Rearranging: n e i(q+g) r e i(q+g) r n ω ω j ɛ = δ = ɛ n + n e i(q+g) r e i(q+g) r n ω ω j δ n e i(q+g) r e i(q+g) r n (ɛ ɛ n) ω ω j ɛ /S

19 Getting δ Starting from the definition: S Rearranging: n e i(q+g) r e i(q+g) r n ω ω j ɛ = δ = ɛ n + n e i(q+g) r e i(q+g) r n ω ω j δ n e i(q+g) r e i(q+g) r n (ɛ ɛ n) ω ω j ɛ /S The ɛ i are eigenvalues of Ĥ with eigenstates i : δ = ɛ n + n e i(q+g) r [Ĥ(r ), e i(q+g) r ] n /S [ω ω j ɛ ]

20 Getting δ Starting from the definition: S Rearranging: n e i(q+g) r e i(q+g) r n ω ω j ɛ = δ = ɛ n + n e i(q+g) r e i(q+g) r n ω ω j δ n e i(q+g) r e i(q+g) r n (ɛ ɛ n) ω ω j ɛ /S The ɛ i are eigenvalues of Ĥ with eigenstates i : δ = ɛ n + n e i(q+g) r [Ĥ(r ), e i(q+g) r ] n /S [ω ω j ɛ ] Working out the ommutator: we obtain δ = ɛ n + [Ĥ(r), e i(q+g) r ] = e i(q+g) r [ q + G 2 q + G ] + (q + G) i n e i(q+g) r e i(q+g) r [i ] n (q + G) /S [ω ω j ɛ ]

21 The Polarizability We an apply a similar approah to the polarizability o χ 0 GG (q, ω) = v v e i(q+g) r e i(q+g ) r v ω (ɛ ɛ v ) + iη + A.R.

22 The Polarizability We an apply a similar approah to the polarizability o χ 0 GG (q, ω) = This an be rewritten as: χ 0 GG (q, ω) = v v v e i(q+g) r e i(q+g ) r v ω (ɛ ɛ v ) + iη v e ( i(q+g) r ) e i(q+g ) r v ω (δ v (q, G, G, ω) ɛ v ) + iη = + A.R. + A.R.

23 Band Struture of Solid Argon: G 0 W 0 Blak: SOS; Red: EET; Blue: LDA

24 SnO 2 : Band gap LDA G 0 W 0 G 0 W 0 Experiment (SOS) (EET) SnO 2 (E g ) SOS: 1600 bands (Σ) EET: 34 valene bands

25 SnO 2 : Band gap LDA G 0 W 0 G 0 W 0 Experiment (SOS) (EET) SnO 2 (E g ) SOS: 1600 bands (Σ) EET: 34 valene bands - Good agreement between SOS approah and EET. - G 0 W 0 band gaps not in agreement with experiment.

26 SnO 2 : Band gap LDA G 0 W 0 G 0 W 0 Experiment (SOS) (EET) SnO 2 (E g ) SOS: 1600 bands (Σ) EET: 34 valene bands - Good agreement between SOS approah and EET. - G 0 W 0 band gaps not in agreement with experiment. Inlude self-onsisteny: GW = self-onsistent COHSEX + G 0 W 0 (Bruneval et al. PRB (2006)) LDA GW (EET) Experiment SnO 2 (E g )

27 SnO 2 : Band struture Blak: LDA; Red: GW (EET )

28 Conlusions and Outlook With the EET we an perform GW alulations with oupied states only. The approah is simple with immediate speed ups of at least an order of magnitude for any system size. Our approah is general and an be applied to any spetral representation. EET for GW implemented in ABINIT 6.4 Potential future appliations inlude: - Self-onsisteny beyond COHSEX: updating only oupied states. - RPA total energies - optimized effetive potentials and kernels. JAB, L. Reining, F. Sottile, Phys. Rev. B 82, (R) (2010)

29 SnO 2 : Absorption tail

Electronic Electr onic and and La t La t t ice ice Polar a i r zat za ion n Effe f c e t c s on the the Band Band Structur

Electronic Electr onic and and La t La t t ice ice Polar a i r zat za ion n Effe f c e t c s on the the Band Band Structur Electronic and Lattice Polarization Effects on the Band Structure of Delafossite Transparent Conductive Oxides Fabio Trani 1 IDEA Virtual Lab (In Silico Development for emerging applications) Scuola Normale