Neutral Electronic Excitations:

Transcription

1 Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra Claudio Attaccalite Second Les Houches school in computational physics: ab-initio simulations in condensed matter

2 Motivations: +- hν Absorption Spectroscopy Many Body Effects!!!

3 Motivations(II):Absorption Spectroscopy Absorption linearly related to the Imaginary part of the MACROSCOPIC dielectric constant (frequency dependent)

4 Outline Response of the system to a perturbation Linear Response Regime How can we calculate the response of the system? Time Dependent DFT and Bethe Salpeter Equation Some applications and recent steps forward Conclusions

5 Spectroscopy

6 Theoretical Spectroscopy Propagation i Ψ=H +V ext (r,t )Ψ t1 [ Schrondinger eq. ] i + ei +V ext Gij (t 1, t 2 )=δ(t 1, t 2)+ Σ G Green's functions t1 i ρ(t )=[ H +V ext, ρ(t )] t HARD Correlation Density Matrix 2 i φ=(v h +V xc +V ext )φ +1 / 2 [ p+ A ( j) ] φ Current-DFT t i φ=(t +V h +V xc +V ext ) φ t TD-DFT 1 r r ' Σ(t 1, t 2) 2 3. ρ (r, r, r, r), ρ... V xc, A xc V xc

7 Linear Response Regime (I) The external potential induces a (time-dependent) density perturbation Kubo Formula (1957) ind δ ρ (r, t) ' ' χ ( r t, r t )= = i [δ ρ(r,t )δ ρ(r ' t ' )] δ φ ext (r ', t ' )

8 Linear Response Regime (II) The induced charge density results in a total potential via the V tot ( r t )=V ext ( r t )+ dt ' d r ' v ( r r ' )ρind ( r ' t ' ) Poisson equation. δ ρ(r,t ) δρ(r,t ) δ V tot (r ' ',t ' ') χ (r,r ', t t ')= = δv ext (r ',t ' ) δ V tot (r ' ', t ' ') δ V ext (r ', t ') Kubo Formula χ ( r t, r ' t ' )=χ 0 ( r t, r ' t ' )+ dt 1 dt 2 d r 1 d r 2 χ 0 ( r t, r 1 t 1) v ( r 1 r 2 ) χ ( r 2 t 2, r ' t ' ) ρind V ind ' χ 0 ( r, r )= V tot δ ρind ( r, t) Variation of the charge V tot ( r ' t ' ) Screening of the density w.r.t. The external perturbation total potential.

9 Linear Response Regime (III) The screening is described by the inverse of the microscopic δ V tot ( r t ) ϵ ( r t, r t )= δv ext ( r t ) 1 dielectric function ' ' =δ ( r r ' )+ dt ' ' d r ' ' v ( r r ' ' )χ ( r ' ', r ' ) Twofold physical meaning : Microscopic level: screening of the interaction between charge carriers in the system In the long wave length limit it determines the macroscopic dielectric function which gives rise to screening of the external perturbation The convolution integrals in real space can be reduced to products is Fourier space ϵ 1 G G ' (q, ω)=1+ v G (q)χ G G ' (q, ω) G=G '=0

10 Optical Absorption: Time Dependent DFT 1 2 [ + V eff (r, t)] ψi (r,t )=i ψi (r, t) 2 t N ρ(r,t )= ψi (r,t ) 2 i=1 V eff (r,t )=V H (r, t)+ V xc (r, t)+ V ext (r, t) Interacting System Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996) δρi χ= δ V ext δρni χ0= δv eff... by using... δρ I =δρni χ δ V ext =χ 0 (δ V ext + δ V H + δv xc ) δ V H δ V xc χ=χ (1+ + ) δ V ext δ V ext 0 vχ Non Interacting System TDDFT is an exact f xc χ theory for neutral excitations! χ (q, ω)=χ 0 (q, ω)+ χ 0 (q, ω)(v+ f xc (q,ω)) χ (q,ω)

11 Why does paper turn yellow? Treasure map By comparing ultraviolet-visible reflectance spectra of ancient and artificially aged modern papers with ab- initio TD-DFT calculations, it was possible to identify and estimate the abundance of oxidized functional groups acting as chromophores and responsible of paper yellowing. yellowing A. Mosca Conte et al., Phys. Rev. Lett. 108, (2012)

12 Optical Absorption: Microscopic View (II) Elementary process of absorption: Photon creates a single e-h pair W= h 2π 2 ϕ e v ϕ δ(ϵi ϵ j ℏ ω) ℑϵ(ω) i j ℏ i, j Non Interacting Non Interacting Particles quasi-particles ϵi, ϵ j GW corrected ϵi, ϵ j Hartree, HF, DFT e Independent transitions energies

13 Optical Absorption: Microscopic View (III) Direct and indirect interactions between an e-h pair created by a photon Summing up all such interaction processes we get: L(r 1 t 1 ; r 2 t 2 ; r 3 t 3 ; r 4 t 4 )=L(1,2,3,4) The equation for L is the Bethe Salpeter Equation. The poles are the neutral excitations.

14 Derivation of the Bethe-Salpeter equation (1) What we want: δ V (1) ϵ (1,2)= δ U (2) 1 i=r i, t i... by using... V (1)=U (1) i ℏ d3 v (1,3)ρ(3) δ ρ(3) ϵ (1,2)=δ(1,2)+ d3 v (1,3) δ U (2) 1 The density is related to the Green's function by... by the identity... ρ(1) = i ℏ G(1,1+ ) δ G (1,2) G2 (1,3 ; 2,3 )=G(1,2)G(3,3 ) δ U (3) + + Reducible polarizability δ ρ(1) χ (1,2)= =i ℏ [G2 (1,2 ; 1+,2 + ) G(1,1+ )G(2,2+ )] δ U (2) χ (1,2)= i ℏ L(1,2 ; 1+,2+ ) two-particle correlation function G. Strinati, Rivista del Nuovo Cimento, 11, 1 (1988)

15 Derivation of the Bethe-Salpeter equation (2) What we have: [i ℏ h(1) U (1)]G(1,2) d4 Σ(3,4)G (4,2)=δ(1,2) t δ G (1,1+ ) δ ρ(1) χ (1,2)= i = = δ ρ(1)δ ρ(2) δ U (2) δ U (2) Using: δ G(1,4) δ G 1 (2,3) =L(1,5,4,6)= G(1,2) G(3,4) δ U (5,6) δ U (5,6) G 1 (1,2)=G(0) 1 (1,2) U (1)δ(1,2) Σ(1,2) Just the Dyson equation for G-1 Dyson equation

16 Derivation of the Bethe-Salpeter equation (3) L=L0+ L0 [ v+ δ Σ ] L δg Bethe-Salpeter Equation! 0 L (1,2,3,4)=G(1,4)G(2,3) Coulomb term Σ(1, 2)=G(1,2) v (2,1) => Screened Coulomb term Σ GW (1,2)= ig (1,2)W (2,1) Time-Dependent Hartree-Fock => Standard Bethe-Salpeter equation (Time-Dependent Screened Hartree-Fock) δ(g W ) L= L0 + L0 [ v ]L δg

17 Feynman's diagrams and Bethe-Salpeter equation L= L0 + L0 [ v W ] L L(1234)=L0 (1234)+ + L0 (1256)[v (57)δ (56)δ(78) W (56)δ(57)δ (68)] L(7834) = + Quasihole and quasielectron Intrinsc 4-point equation. It describes the (coupled) progation of two particles, the electron and the hole! Retardation effects are neglected W (1,2)=W (r 1, r 2 )δ(t 1,t 2 ) L(1,2,3,4)=L(r 1, r 2, r 3, r 4 ; t t 0)=L(1,2,3,4, ω)

18 Bethe-Salpeter equation (4-points - space and time) L(1,2,3,4)=L(r 1, r 2, r 3, r 4 ; t t 0)=L(1,2,3,4, ω) Should we invert the equation for L for each frequency??? H exc (n1 n2 ),(n3 n4 ) A (n3 n 4 ) λ =E λ A (n1 n2 ) λ We work in transition space...

19 Effective two particle Hamiltonian It corresponds to transitions at Pseudo-Hermitian positive absorption frequencies v. Tamm Dancoff!!! It corresponds to transitions at negative absorption frequencies v. ϵ M (ω)=1 lim v (q) q 0 v k q e i q r c k 2 λ vc, k E λ ω i η

20 Bethe Salpeter Equation Historical remarks First solution of BSE with dynamical effects: Shindo approximation JPSJ 29, 278(1970) Plane-waves implementation G. Onida et al. PRL 75, 818 (1995) 1974 First applications in solids: W. Hanke and L.J. Sham PRL 33, 582(1974) G. Strinati, H.J. Mattausch and W. Hanke PRL 45, 290 (1980)

21 Some results Bruneval et al., PRL 97, (2006) Strinati et al., Rivista del Nuovo Cimento 11, 1 (1988) Albrecht et al., PRL 80, 4510 (1998) Bruno et al., PRL 98, (2007) Tiago et al., PRB 70, (2004) V. Garbuio et al., PRL 97, (2006)

22 Excitons in nanoscale systems Frenkel excitons in photosynthesis Nanotubes/Nanowires Colloidal quantum dots Excitons in nanoscale systems Gregory D. Scholes, Garry Rumbles Nature Materials 5, (2006)

23 ... advances...

24 Beyond Tamm-Dancoff approximation! Mixed excitonic-plasmonic excitations in nanostructures (Nanoletters, 6, 257(2010)) Excited states of biological chromophores (J. Chem. Theory Comput., 6, (2010))

25 Ab-initio broadening in BSE Ab-Initio finite temperature excitons A. Marini PRL 101, (2008). Ab Initio Calculation of Optical Spectra of Liquids: Many-Body Effects in the Electronic Excitations of Water V. Garbuio et al., PRL 97, (2006).

26 Dynamical Excitonic Effects in Metals and Semiconductors The inclusion of the full dynamic screening in the BS equation complicates its numerical solution tremendously, but it is possible to perform an expansion in the dynamical part of the screened interaction. First solution of this problem the so-called Shindo approximation (J. Phys. Soc. Jpn. 29, 278(1970)) Dynamical effects in Sodium clusters Dynamical effects in metals and semiconductors A. Marini and R. Del sole PRL, 91, (2003). G. Pal et al. EPJ B 79, 327 (2011)

27 Non-linear response: frequency and time domain Second-order response Bethe-Salpeter equation (PRA, 83, (2011)) Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equation (PRB, 84, (2011))

28 References!!! Reviews: Application of the Green s functions method to the study of the optical properties of semiconductors Nuovo Cimento, vol 11, pg 1, (1988) G. Strinati Effects of the Electron Hole Interaction on the Optical Properties of Materials: the Bethe Salpeter Equation Physica Scripta, vol 109, pg 141, (2004) G. Bussi Electronic excitations: density-functional versus many-body Green's-function approaches RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio Books: On the web:

29 DFT meets Many-Body 29

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35 .. with some algebra...

36 References!!! Reviews: Application of the Green s functions method to the study of the optical properties of semiconductors Nuovo Cimento, vol 11, pg 1, (1988) G. Strinati Effects of the Electron Hole Interaction on the Optical Properties of Materials: the Bethe Salpeter Equation Physica Scripta, vol 109, pg 141, (2004) G. Bussi Electronic excitations: density-functional versus many-body Green's-function approaches RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio Books: On the web:

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38 Optical Absorption: Microscopic Limit δ ρni =χ 0 δ V tot 0 χ = ij ϕi (r) ϕ*j (r) ϕ*i (r ' ) ϕj (r ' ) ω (ϵi ϵ j )+ i η Hartree, Hartree-Fock, dft... Non Interacting System Absorption by independent Kohn-Sham particles =ℑ χ 0 = j D i 2 δ(ω (ϵ j ϵi )) ij 2 8π ϵ (ω)= 2 ω Particles are interacting! '' 2 ϕ e v ϕ δ (ϵi ϵ j ℏ ω) i j i, j