BSE and TDDFT at work

Transcription

1 BSE and TDDFT at work Claudio Attaccalite CECAM Yambo School 2013 (Lausanne)

2 Optical Absorption: Microscopic View 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.

3 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)

4 Feynman's diagrams and Bethe-Salpeter equation L= L0 + L0 [ v W ] L L(1234)=L0 (1234)+ L [v W ] L 7834 = + Quasihole and quasielectron Intrinsic 4-point equation. It describes the (coupled) propagation of two particles, the electron and the hole! Retardation effects are neglected W 1,2 =W r 1, r 2 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,

5 Construction of the BSE 1/2 We expand L in the independent particle basis L(r 1, r 2, r ' 1, r ' 2 ω)= k 1, k2, k 3, k 4 ϕk (r 1 )ϕ k (r 2 ) L k 1 2 1, k2, k 3, k 4 ϕk (r ' 1 ) ϕ k (r ' 2 ) 3 4 We start from L0 L0 (r 1, r 2, r ' 1, r ' 2 ω)=2 d ω' G0 (r 1, r ' 2 ; ω+ω' )G0 (r 2, r ' 1 ; ω' ) 2π where ϕ k,i (r 1 )ϕ k, i (r 2 ) ℏ ω ϵk, i +i ηsign(ϵk, i ϵ F ) G0 (r 1, r 2 ; ω)= k,i...integrating in the frequency we get... L 0 k1, k2, k ' 1, k ' 2 =δ k 1, k2 δk ' 2i f (ϵ k ) f (ϵ k ' ) ℏ ϵ k ' ϵk +i δ+ ℏ ω 2 1, k'

6 Construction of the BSE 2/2 Time-dependent Hartree term: e h e h H TD harteee =2 ϕ (r )ϕ (r )V (r r ')ϕ (r ') ϕ ij, kl i j k l (r ') Screened Exchange Coulomb term:e h e h H TD SEX = ϕ (r )ϕ (r ' )W (r, r ')ϕ (r ) ϕ ij, kl i j k l (r ')...and now let's solve the equation...

7 Bethe-Salpeter equation (4-points - space and time) Should we invert the equation L 1,2,3,4 =L r 1, r 2, r 3, r 4 ; t t 0 =L 1,2,3,4, for L for each frequency??? The frequency term can be separated and an-effective Hamiltonian can be derived without any frequency dependency L=L0 + L 0 [v W ] L 1 1 L +[v W ] = L [ 0 ] H exc n1 n2, n3 n4 A n3 n4 =E A n1 n2 We work in transition space... Original BSE We solve the inverse Bethe-Salpeter eq., because it is easier

8 How to transform the BSE in an eigenvalues problem Using the definition of L0 That is diagonal in the e/h space

9 We can solve the equation once for all frequency!!!!...with some linear algebra...

10 Absoprtion spectra and BSE

11 BSE calculation in practice

12 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)

13 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)

14 BSE for charge transfer excitons donor-acceptor complexes: benzene, naphthalene, and anthracene derivatives with the tetracyanoethylene acceptor X. Blase and C. Attaccalite Apl. Phys. Lett. 99, (2011)

15 Exciton analysis

16 Solving the equations in a smart way...

17 The BSE can be large... too large R

18 Tamm-Dancoff approximation

19 The dielectric constant doesn't require too much information

20 Let's come back to the original formula we can write the dielectric constant as [ ϵ 2 (ω, q )=4 π ℑ P 1 ω H EXC +i η lim q 0 e iqr P = 0 q...and ask the help of mathematicians... P ]

21 Lanczos-Haydock method

22 Lanczos-Haydock algorithm

23 Lanczos-Haydock performance

24 What about TDDFT?

25 TDDFT versus BSE BSE L(1234)=L0 (1234)+ + L0 (1256)[ v (57) δ(56) δ(78) W (56) δ(57)δ (68)] L(7834) TDDFT χ (12)=χ 0 (12)+ χ 0 (13)[v (34)+f xc ]χ ( 42) BSE is a 4-points equation => unavoidable TDDFT is a 2-points equation => that can be rewritten as a 4-point equation

26 TDDFT in G-space Simple static fxc case: χ G, G ' ( q, ω)=χ 0G, G ' (q,ω)+χ 0G, G (q,ω)( vg ( q)+ f Gxc G (q)) χ G 2 2 2, 3 3,G ( q,ω) Microscopic dielectric constant: ϵ 1 G, G ' ( q,ω)=δ G, G ' + v G (q)χ G, G ' (q, ω) Macroscopic dielectric constant: M ϵ (q,ω)= 1 ϵ 1 G=0, G '=0 ( q, ω) Advantages: 2-points eq. Disadvantages: the eqs. Has to be solved for each frequency

27 TDDFT in e/h space Time-dependent Hartree term: e h e h H TD hartree =2 ϕ (r )ϕ (r )V (r r ' )ϕ (r ')ϕ ij, kl i j k l (r ' ) Time-dependent exchange correlation function: H TD EXC ij, kl V xc (r ) = ϕ (r )ϕ (r )f xc (r, r ')ϕ (r ' )ϕ (r ' ) f xc (r, r ')= ρ(r ' ) e i h j e k h l e h e h H TD SEX = ϕ (r )ϕ (r ' )W (r, r ') ϕ (r ) ϕ i ij, kl j k l (r ' ) BSE

28 Beyond the TammDancoff approx.

29 Tamm-Dancoff breakdown 1

30 Tamm-Dancoff breakdown 2

31 Don't worry! Haydock method still works

32 SUMMARY Optical spectra can be calculated by mean of Green's function theory TDDFT and BSE can efficiently be - formulated in the e/h space By using Lanczos-Haydock approach we do not need to diagonalize the full matrix!

33 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: