Introduction to manybody Green s functions


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1 Introduction to manybody Green s functions Matteo Gatti European Theoretical Spectroscopy Facility (ETSF) NanoBio Spectroscopy Group  UPV San Sebastián  Spain ELK school  CECAM 2011
2 Outline 1 Motivation 2 Oneparticle Green s functions: GW approximation 3 Twoparticle Green s functions: BetheSalpeter equation 4 Micromacro connection
3 References Francesco Sottile PhD thesis, Ecole Polytechnique (2003) francesco/tesi_dot.pdf Fabien Bruneval PhD thesis, Ecole Polytechnique (2005) bruneval_these.pdf Giovanni Onida, Lucia Reining, and Angel Rubio Rev. Mod. Phys. 74, 601 (2002). G. Strinati Rivista del Nuovo Cimento 11, (12)1 (1988).
4 Outline 1 Motivation 2 Oneparticle Green s functions: GW approximation 3 Twoparticle Green s functions: BetheSalpeter equation 4 Micromacro connection
5 Motivation Theoretical spectroscopy Calculate and reproduce Understand and explain Predict Exp. at 30 K from: P. Lautenschlager et al., Phys. Rev. B 36, 4821 (1987).
6 Theoretical Spectroscopy Which kind of spectra? Which kind of tools?
7 Why do we have to study more than DFT? Absorption spectrum of bulk silicon in DFT How can we understand this?
8 Why do we have to study more than DFT? Absorption spectrum of bulk silicon in DFT Spectroscopy is exciting!
9 MBPT vs. TDDFT: different worlds, same physics MBPT based on Green s functions oneparticle G: electron addition and removal  GW twoparticle L: electronhole excitation  BSE moves (quasi)particles around is intuitive (easy) TDDFT based on the density response function χ: neutral excitations moves density around is efficient (simple)
10 Response functions External perturbation V ext applied on the sample V tot acting on the electronic system Potentials Dielectric function δv tot = δv ext + δv ind ɛ = δv ext δv tot δv ind = vδρ = 1 v δρ ɛ 1 = δv tot = 1 + v δv ext δv tot δρ δv ext
11 Response functions External perturbation V ext applied on the sample V tot acting on the electronic system Dielectric function P = ɛ = δv ext δv tot = 1 vp ɛ 1 = δv tot δv ext = 1 + vχ δρ δv tot χ = δρ δv ext χ = P + Pvχ P = χ 0 + χ 0 f xc P
12 Micromacro connection MicroscopicMacroscopic connection: local fields χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω)v G1 (q)χ G1,G (q, ω) ɛ 1 G,G (q, ω) = δ G,G + v G (q)χ G,G (q, ω) ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Adler, Phys. Rev. 126 (1962); Wiser, Phys. Rev. 129 (1963).
13 Micromacro connection MicroscopicMacroscopic connection: local fields ɛ M (q, ω) = 1 v G=0 (q) χ G=0,G =0(q, ω) χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω) v G1 (q) χ G1,G (q, ω) v G (q) = 0 for G = 0 v G (q) = v G (q) for G 0 Hanke, Adv. Phys. 27 (1978).
14 Absorption spectra Absorption spectra Abs(ω) = lim q 0 Imɛ M (q, ω) Abs(ω) = lim q 0 Im [v G=0 (q) χ G=0,G =0(q, ω)] Absorption response to V ext + V macro ind
15 Independent particles: KohnSham Independent transitions: ɛ 2 (ω) = 8π2 ϕ Ωω 2 j e v ϕ i 2 δ(ε j ε i ω) ij
16 What is an electron?
17 Outline 1 Motivation 2 Oneparticle Green s functions: GW approximation 3 Twoparticle Green s functions: BetheSalpeter equation 4 Micromacro connection
18 Photoemission Direct Photoemission Inverse Photoemission
19 Why do we have to study more than DFT? adapted from M. van Schilfgaarde et al., PRL 96 (2006).
20 Oneparticle Green s function The oneparticle Green s function G Definition and meaning of G: ig(x 1, t 1 ; x 2, t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N for for t 1 > t 2 ig(x 1, t 1 ; x 2, t 2 ) = N ψ(x 1, t 1 )ψ (x 2, t 2 ) N t 1 < t 2 ig(x 1, t 1 ; x 2, t 2 ) = N ψ (x 2, t 2 )ψ(x 1, t 1 ) N
21 Oneparticle Green s function t 1 > t 2 N ψ(x 1, t 1 )ψ (x 2, t 2 ) N t 1 < t 2 N ψ (x 2, t 2 )ψ(x 1, t 1 ) N
22 Oneparticle Green s function What is G? Definition and meaning of G: [ ] G(x 1, t 1 ; x 2, t 2 ) = i < N T ψ(x 1, t 1 )ψ (x 2, t 2 ) N > Insert a complete set of N + 1 or N 1particle states. This yields G(x 1, t 1 ; x 2, t 2 ) = i j f j (x 1 )f j (x 2 )e iε j (t 1 t 2 ) [θ(t 1 t 2 )θ(ε j µ) θ(t 2 t 1 )Θ(µ ε j )]; where: ε j = E(N + 1, j) E(N), ε j > µ E(N) E(N 1, j), ε j < µ f j (x 1 ) = N ψ (x 1) N + 1, j, ε j > µ N 1, j ψ (x 1 ) N, ε j < µ
23 Oneparticle Green s function What is G?  Fourier transform G(x, x, ω) = j Fourier Transform: f j (x)f j (x ) ω ε j + iηsgn(ε j µ). Spectral function: A(x, x ; ω) = 1 π ImG(x, x ; ω) = j f j (x)f j (x )δ(ω ε j ).
24 Photoemission Direct Photoemission Inverse Photoemission Oneparticle excitations poles of oneparticle Green s function G
25 Oneparticle Green s function Oneparticle Green s function From oneparticle G we can obtain: oneparticle excitation spectra groundstate expectation value of any oneparticle operator: e.g. density ρ or density matrix γ: ρ(r, t) = ig(r, r, t, t + ) γ(r, r, t) = ig(r, r, t, t + ) groundstate total energy
26 Oneparticle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Timeindependent perturbation theories: messy. Textbooks: adiabatically switched on interaction, GellMannLow theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...
27 Oneparticle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Timeindependent perturbation theories: messy. Textbooks: adiabatically switched on interaction, GellMannLow theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...
28 Oneparticle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Timeindependent perturbation theories: messy. Textbooks: adiabatically switched on interaction, GellMannLow theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...
29 Functional approach to the MB problem Equation of motion To determine the 1particle Green s function: [ ] i h 0 (1) G(1, 2) = δ(1, 2) i t 1 d3v(1, 3)G 2 (1, 3, 2, 3 + ) Do the Fourier transform in frequency space: [ω h 0 ]G(ω) + i vg 2 (ω) = 1 where h 0 = v ext is the independent particle Hamiltonian. The 2particle Green s function describes the motion of 2 particles.
30 Unfortunately, hierarchy of equations G 1 (1, 2) G 2 (1, 2; 3, 4) G 2 (1, 2; 3, 4) G 3 (1, 2, 3; 4, 5, 6)...
31 Selfenergy Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G 0 (ω) that corresponds to the Hamiltonian h 0 Everything that is unknown is put in Σ(ω) = G 1 0 (ω) G 1 (ω) This is the definition of the selfenergy Thus, [ω h 0 ]G(ω) Σ(ω)G(ω) = 1 to be compared with [ω h 0 ]G(ω) + i vg 2 (ω) = 1
32 Selfenergy Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G 0 (ω) that corresponds to the Hamiltonian h 0 Everything that is unknown is put in Σ(ω) = G 1 0 (ω) G 1 (ω) This is the definition of the selfenergy Thus, [ω h 0 ]G(ω) Σ(ω)G(ω) = 1 to be compared with [ω h 0 ]G(ω) + i vg 2 (ω) = 1
33 Oneparticle Green s function Trick due to Schwinger (1951): introduce a small external potential U(3), that will be made equal to zero at the end, and calculate the variations of G with respect to U δg(1, 2) δu(3) = G 2 (1, 3; 2, 3) + G(1, 2)G(3, 3).
34 Hedin s equation Hedin s equations Σ =igw Γ G =G 0 + G 0 ΣG Γ =1 + δσ δg GGΓ P = iggγ W =v + vpw L. Hedin, Phys. Rev. 139 (1965)
35 GW bandstructure: photoemission additional charge
36 GW bandstructure: photoemission additional charge reaction: polarization, screening GW approximation 1 polarization made of noninteracting electronhole pairs (RPA) 2 classical (Hartree) interaction between additional charge and polarization charge
37 Hedin s equation and GW GW approximation Σ =igw Γ G =G 0 + G 0 ΣG Γ =1 P = iggγ W =v + vpw L. Hedin, Phys. Rev. 139 (1965)
38 Hedin s equation and GW GW approximation Σ =igw G =G 0 + G 0 ΣG Γ =1 P = igg W =v + vpw L. Hedin, Phys. Rev. 139 (1965)
39 GW corrections Standard perturbative G 0 W 0 H 0 (r)φ i (r) + H 0 (r)ϕ i (r) + V xc (r)ϕ i (r) = ɛ i ϕ i (r) dr Σ(r, r, ω = E i ) φ i (r ) = E i φ i (r) Firstorder perturbative corrections with Σ = igw : E i ɛ i = ϕ i Σ V xc ϕ i Hybersten and Louie, PRB 34 (1986); Godby, Schlüter and Sham, PRB 37 (1988)
40 GW results M. van Schilfgaarde et al., PRL 96 (2006).
41 Independent (quasi)particles: GW Independent transitions: ɛ 2 (ω) = 8π2 ϕ Ωω 2 j e v ϕ i 2 δ(e j E i ω) ij
42 What is wrong? What is missing?
43 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
44 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
45 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
46 Outline 1 Motivation 2 Oneparticle Green s functions: GW approximation 3 Twoparticle Green s functions: BetheSalpeter equation 4 Micromacro connection
47 Beyond RPA P(12) = ig(12)g(21) = P 0 (12) Independent particles (RPA)
48 Beyond RPA P(12) = ig(13)g(42)γ(342) Interacting particles (excitonic effects)
49 From Hedin s equations to BSE From Hedin... P = iggγ Γ = 1 + δσ δg GGΓ
50 From Hedin s equations to BSE From Hedin......to BetheSalpeter P = iggγ Γ = 1 + δσ δg GGΓ ( δσ ) L = L 0 + L 0 v + i L δg
51 The BetheSalpeter equation Exercise Formal derivation δg(12) L(1234) = i δv = +ig(15) δg 1 (56) ext(34) δv G(62) ext(34) = + ig(15) δ[g 1 0 (56) Vext(56) Σ(56)] G(62) δv ext(34) [ δvh (5)δ(56) = ig(13)g(42) + ig(15)g(62) δσ(56) ] δv ext(34) δv ext(34) [ δvh (5)δ(56) = ig(13)g(42) + ig(15)g(62) δσ(56) ] δg(78) δg(78) δg(78) δv ext(34) [ L(1234) =L 0 (1234) + L 0 (1256) v(57)δ(56)δ(78) + i δσ(56) δg(78) ] L(7834)
52 The BetheSalpeter equation [ L(1234) = L 0 (1234) + L 0 (1256) v(57)δ(56)δ(78) + i δσ(56) δg(78) ] L(7834) Polarizabilities δg(12) L(1234) = i δv ext (34) χ(12) = δρ(1) δv ext (2) L(1122) = χ(12)
53 The BetheSalpeter equation Approximations ( δσ ) L = L 0 + L 0 v + i L δg
54 The BetheSalpeter equation Approximations Σ igw ( δσ ) L = L 0 + L 0 v + i L δg Approximation:
55 The BetheSalpeter equation Approximations Σ igw ( δ(gw )) L = L 0 + L 0 v L δg Approximation: δ(gw ) δg = W + GδW δg W
56 The BetheSalpeter equation Approximations Final result: L = L 0 + L 0 (v W )L
57 The BetheSalpeter equation BetheSalpeter equation L(1234) = L 0 (1234)+ L 0 (1256)[v(57)δ(56)δ(78) W (56)δ(57)δ(68)]L(7834)
58 Absorption spectra in BSE Bulk silicon G. Onida, L. Reining, and A. Rubio, RMP 74 (2002).
59 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[v(57)δ(56)δ(78) W (56)δ(57)δ(68)]L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)
60 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[ v(57)δ(56)δ(78) W (56)δ(57)δ(68)] L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)
61 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[ v(57)δ(56)δ(78) W (56)δ(57)δ(68)] L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)
62 Solving BSE Dielectric function L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) [ ɛ M (ω) = 1 lim v G=0 (q) q 0 ] drdr e iq(r r ) L(r, r, r, r, ω)
63 Solving BSE L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) Transition space How to solve it? L (n1 n 2 )(n 3 n 4 )(ω) = φ n 1 (r 1 )φ n2 (r 2 ) L(r 1 r 2 r 3 r 4 ω) φ n 3 (r 3 )φ n4 (r 4 ) = L
64 Solving BSE L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) Transition space How to solve it? L (n1 n 2 )(n 3 n 4 )(ω) = φ n 1 (r 1 )φ n2 (r 2 ) L(r 1 r 2 r 3 r 4 ω) φ n 3 (r 3 )φ n4 (r 4 ) = L
65 Exercise Calculate: L 0 (r 1, r 2, r 3, r 4, ω) = ij L 0 = (f j f i ) φ i (r 1)φ j (r 2 )φ i (r 3 )φ j (r 4) ω (E i E j ) f n1 f n2 ω (E n2 E n1 ) δ n 1 n 3 δ n2 n 4
66 Solving BSE BSE in transition space We consider only resonant optical transitions for a nonmetallic system: (n 1 n 2 ) = (vkck) (vc) L = L 0 + L 0 ( v W ) L L = [1 L 0 ( v W )] 1 L 0 L = [L 1 0 ( v W )] 1 L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v )
67 Solving BSE L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v ) Spectral representation of a hermitian operator [H exc ωi ] 1 = λ L (vc)(v c )(ω) = λ H exc A λ = E λ A λ A λ A λ E λ ω A (vc) λ A (v c ) λ E λ ω (f c f v )
68 Solving BSE L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v ) L [H exc ωi ] 1 Spectral representation of a hermitian operator [H exc ωi ] 1 = λ L (vc)(v c )(ω) = λ H exc A λ = E λ A λ A λ A λ E λ ω A (vc) λ A (v c ) λ E λ ω (f c f v )
69 Absorption spectra in BSE Independent (quasi)particles Abs(ω) vc v D c 2 δ(e c E v ω) Excitonic effects [H el + H hole +H el hole ] A λ = E λ A λ Abs(ω) λ vc A (vc) λ v D c 2 δ(e λ ω) mixing of transitions: v D c 2 vc A(vc) λ v D c 2 modification of excitation energies: E c E v E λ
70 BSE calculations A threestep method 1 LDA calculation KohnSham wavefunctions ϕ i 2 GW calculation GW energies E i and screened Coulomb interaction W 3 BSE calculation solution of H exc A λ = E λ A λ with: H (vc)(v c ) exc = (E c E v )δ vv δ cc + (f v f c ) vc v W v c excitonic eigenstates A λ, E λ spectra ɛ M (ω)
71 A bit of history derivation of the equation (bound state of deuteron) E. E. Salpeter and H. A. Bethe, PR 84, 1232 (1951). BSE for exciton calculations L.J. Sham and T.M. Rice, PR 144, 708 (1966). W. Hanke and L. J. Sham, PRL 43, 387 (1979). first ab initio calculation G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, PRL 75, 818 (1995). first ab initio calculations in extended systems S. Albrecht, L. Reining, R. Del Sole, and G. Onida, PRL 80, 4510 (1998). L. X. Benedict, E. L. Shirley, and R. B. Bohn, PRL 80, 4514 (1998). M. Rohlfing and S. G. Louie, PRL 81, 2312 (1998).
72 Continuum excitons Bulk silicon G. Onida, L. Reining, and A. Rubio, RMP 74 (2002).
73 Bound excitons Solid argon F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76 (2007).
74 The Wannier model BetheSalpeter equation H exca λ = E λ A λ H (vc)(v c ) exc = (E c E v)δ vv δ cc + v W Wannier model two parabolic bands E c E v = E g + k 2 2µ 2 2µ no local fields ( v = 0) and effective screened W W (r, r ) = 1 ɛ 0 r r solution = Rydberg series for effective H atom E n = E g R eff n 2 with R eff = R µ ɛ 2 0
75 Exciton analysis Exciton amplitude: Ψ λ (r h, r e ) = vc A (vc) λ φ v(r h )φ c (r e ) Graphene nanoribbon Manganese Oxide D. Prezzi, et al., PRB 77 (2008). C. Rödl, et al., PRB 77 (2008).
76 Outline 1 Motivation 2 Oneparticle Green s functions: GW approximation 3 Twoparticle Green s functions: BetheSalpeter equation 4 Micromacro connection
77 Micromacro connection Observation At long wavelength, external fields are slowly varying over the unit cell: dimension of the unit cell for silicon: 0.5 nm visible radiation 400 nm < λ < 800 nm Total and induced fields are rapidly varying: they include the contribution from electrons in all regions of the cell. Large and irregular fluctuations over the atomic scale.
78 Micromacro connection Observation One usually measures quantities that vary on a macroscopic scale. When we calculate microscopic quantities we need to average over distances that are large compared to the cell parameter small compared to the wavelength of the external perturbation. The differences between the microscopic fields and the averaged (macroscopic) fields are called the crystal local fields.
79 Suppose that we are able to calculate the microscopic dielectric function ɛ, how do we obtain the macroscopic dielectric function ɛ M that we measure in experiments?
80 Micromacro connection Fourier transform In a periodic medium, every function V (r, ω) can be represented by the Fourier series V (r, ω) = qg V (q + G, ω)e i(q+g)r or: V (r, ω) = q e iqr G V (q + G, ω)e igr = q e iqr V (q, r, ω) where: V (q, r, ω) = G V (q + G, ω)e igr V (q, r, ω) is periodic with respect to the Bravais lattice and hence is the quantity that one has to average to get the corresponding macroscopic potential V M (q, ω).
81 Micromacro connection Averages V M (q, ω) = 1 Ω c drv (q, r, ω) V (q, r, ω) = G V (q + G, ω)e igr Therefore: V M (q, ω) = G V (q + G, ω) 1 Ω c dre igr = V (q + 0, ω) The macroscopic average V M corresponds to the G = 0 component of the microscopic V. Example V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω)
82 Micromacro connection Averages V M (q, ω) = 1 Ω c drv (q, r, ω) V (q, r, ω) = G V (q + G, ω)e igr Therefore: V M (q, ω) = G V (q + G, ω) 1 Ω c dre igr = V (q + 0, ω) The macroscopic average V M corresponds to the G = 0 component of the microscopic V. Example V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω)
83 Micromacro connection Fourier transforms Fourier transform of a function f (r, r, ω): f (q + G, q + G, ω) = drdr e i(q+g)r f (r, r, ω)e +i(q+g )r f G,G (q, ω) Therefore the relation V tot (r 1, ω) = dr 2 ɛ 1 (r 1, r 2, ω)v ext (r 2, ω) in the Fourier space becomes: V tot (q + G, ω) = G ɛ 1 G,G (q, ω)v ext (q + G, ω)
84 Micromacro connection Fourier transforms Fourier transform of a function f (r, r, ω): f (q + G, q + G, ω) = drdr e i(q+g)r f (r, r, ω)e +i(q+g )r f G,G (q, ω) Therefore the relation V tot (r 1, ω) = dr 2 ɛ 1 (r 1, r 2, ω)v ext (r 2, ω) in the Fourier space becomes: V tot (q + G, ω) = G ɛ 1 G,G (q, ω)v ext (q + G, ω)
85 Micromacro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)
86 Micromacro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)
87 Micromacro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)
88 Micromacro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)
89 Micromacro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)
90 Micromacro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)
91 Micromacro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)
92 Micromacro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)
93 Micromacro connection Spectra ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Abs(ω) = lim q 0 Imɛ M (ω) = lim EELS(ω) = lim Imɛ 1 M q 0 q 0 Im (ω) = lim q 0 Imɛ 1 1 ɛ 1 G=0,G =0 (q, ω) G=0,G =0 (q, ω)
94 Micromacro connection Spectra ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Abs(ω) = lim q 0 Imɛ M (ω) = lim EELS(ω) = lim Imɛ 1 M q 0 q 0 Im (ω) = lim q 0 Imɛ 1 1 ɛ 1 G=0,G =0 (q, ω) G=0,G =0 (q, ω)
95 BSE vs. TDDFT: what in common? BSE L = L 0 + L 0 (v + Ξ)L TDDFT χ = χ 0 + χ 0 (v + f xc )χ
96 The Coulomb term v The Coulomb term v = v 0 + v
97 Local fields reloaded MicroscopicMacroscopic connection: local fields χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω)v G1 (q)χ G1,G (q, ω) G 1 ɛ 1 G,G (q, ω) = δ G,G + v G (q)χ G,G (q, ω) ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Adler, Phys. Rev. 126 (1962); Wiser, Phys. Rev. 129 (1963).
98 Local fields reloaded MicroscopicMacroscopic connection: local fields ɛ M (q, ω) = 1 v G=0 (q) χ G=0,G =0(q, ω) χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω) v G1 (q) χ G1,G (q, ω) G 1 v G (q) = 0 for G = 0 v G (q) = v G (q) for G 0 Hanke, Adv. Phys. 27 (1978).
99 Absorption Abs(ω) = lim q 0 Imɛ M (q, ω) Abs(ω) = lim q 0 Im [v G=0 (q) χ G=0,G =0(q, ω)] χ = P + P v χ EELS Absorption response to V ext + V macro ind Eels(ω) = lim q 0 Im[1/ɛ M (q, ω)] Eels(ω) = lim q 0 Im [v G=0 (q)χ G=0,G =0(q, ω)] χ = P + P(v 0 + v)χ Eels response to V ext
100 The Coulomb term v The Coulomb term v = v 0 + v longrange v 0 difference between Abs and Eels
101 Coulomb term v 0 : Abs vs. Eels F. Sottile, PhD thesis (2003)  Bulk silicon: absorption vs. EELS.
102 The Coulomb term v The Coulomb term v = v 0 + v longrange v 0 difference between Abs and Eels what about v?
103 The Coulomb term v The Coulomb term v = v 0 + v longrange v 0 difference between Abs and Eels what about v? v is responsible for crystal localfield effects
104 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no localfield effects! (equivalent to Fermi s golden rule)
105 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no localfield effects! (equivalent to Fermi s golden rule)
106 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no localfield effects! (equivalent to Fermi s golden rule)
107 Coulomb term v: local fields Bulk silicon: absorption
108 Coulomb term v: local fields A. G. Marinopoulos et al., PRL 89 (2002)  Graphite EELS
109 What are local fields? Effective medium theory Uniform field E 0 applied to a dielectric sphere with dielectric constant ɛ in vacuum. From continuity conditions at the interface: P = 3 ɛ 1 4π ɛ + 2 E 0 Jackson, Classical electrodynamics, Sec. 4.4.
110 What are local fields? Effective medium theory Regular lattice of objects dimensionality d of material ɛ 1 in vacuum MaxwellGarnett formulas dot (O D system) wire (1D system) Imɛ 1 (ω) Imɛ M (ω) 9 [Reɛ 1 (ω) + 2] 2 + [Imɛ 1 (ω)] 2 Imɛ M (ω) Imɛ 1(ω) Imɛ Imɛ 1 (ω) M (ω) 4 [Reɛ 1 (ω) + 1] 2 + [Imɛ 1 (ω)] 2
111 What are local fields? F. Bruneval et al., PRL 94 (2005)  Si nanowires S. Botti et al., PRB 79 (2009)  SiGe nanodots
112 MBPT & TDDFT MBPT helps improving DFT & TDDFT DFT & TDDFT help improving MBPT
113 Conclusion (TD)DFT & MBPT... try to learn both!
114 Many thanks!
115 Acknowledgements Silvana Botti Fabien Bruneval Valerio Olevano Lucia Reining Francesco Sottile Valérie Véniard
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