Introduction to manybody Green s functions


 Teresa Baldwin
 2 years ago
 Views:
Transcription
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
Introduction to Green s functions
Introduction to Green s functions Matteo Gatti ETSF Users Meeting and Training Day Ecole Polytechnique  22 October 2010 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The BetheSalpeter
More informationUniversità degli Studi Roma TRE. Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia
Università degli Studi Roma TRE e Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia Dottorato di Ricerca in Scienze Fisiche della Materia XXIII ciclo Challenges for first principles
More informationOptical Properties of Solids. Claudia AmbroschDraxl Chair of Atomistic Modelling and Design of Materials University Leoben, Austria
Optical Properties of Solids Claudia AmbroschDraxl Chair of Atomistic Modelling and Design of Materials University Leoben, Austria Outline Basics Program Examples Outlook light scattering dielectric tensor
More informationChapter 2. Theoretical framework
Theoretical framework For studying the properties of thin films and surfaces on the atomic scale a wealth of experimental techniques is available. For brevity, we will introduce only those that will be
More informationLecture 3: Optical Properties of Bulk and Nano. 5 nm
Lecture 3: Optical Properties of Bulk and Nano 5 nm The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model) 0 e  n( ) n' n '' n ' = 1 + Nucleus
More informationThe Application of Density Functional Theory in Materials Science
The Application of Density Functional Theory in Materials Science Slide 1 Outline Atomistic Modelling Group at MUL Density Functional Theory Numerical Details HPC Cluster at the MU Leoben Applications
More informationThe Central Equation
The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given
More informationPCV Project: Excitons in Molecular Spectroscopy
PCV Project: Excitons in Molecular Spectroscopy Introduction The concept of excitons was first introduced by Frenkel (1) in 1931 as a general excitation delocalization mechanism to account for the ability
More informationLecture 3: Optical Properties of Bulk and Nano. 5 nm
Lecture 3: Optical Properties of Bulk and Nano 5 nm First H/W#1 is due Sept. 10 Course Info The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model)
More informationSemiconductor Physics
10p PhD Course Semiconductor Physics 18 Lectures NovDec 2011 and Jan Feb 2012 Literature Semiconductor Physics K. Seeger The Physics of Semiconductors Grundmann Basic Semiconductors Physics  Hamaguchi
More information= N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0
Chapter 1 ThomasFermi Theory The ThomasFermi theory provides a functional form for the kinetic energy of a noninteracting electron gas in some known external potential V (r) (usually due to impurities)
More informationExciton dissociation in solar cells:
Exciton dissociation in solar cells: Xiaoyang Zhu Department of Chemistry University of Minnesota, Minneapolis t (fs) 3h! E, k h! Pc Bi e  1 Acknowledgement Organic semiconductors: Mutthias Muntwiler,
More informationQuick and Dirty Introduction to Mott Insulators
Quick and Dirty Introduction to Mott Insulators Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys64/phys64.html
More informationThe electronic and optical properties of conjugated polymers:
The electronic and optical properties of conjugated polymers: predictions from firstprinciples solidstate methods PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,
More informationarxiv:1006.4085v1 [condmat.mtrlsci] 21 Jun 2010
Quasiparticle and Optical Properties of Rutile and Anatase TiO 2 Wei Kang and Mark S. Hybertsen Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973 (Dated: June 22, 21)
More informationWhat is molecular dynamics (MD) simulation and how does it work?
What is molecular dynamics (MD) simulation and how does it work? A lecture for CHM425/525 Fall 2011 The underlying physical laws necessary for the mathematical theory of a large part of physics and the
More informationBasic techniques and tools for the development and maintenance of atomicscale software : the context
CECAM Lyon February 2008 Basic techniques and tools for the development and maintenance of atomicscale software : the context X. Gonze Université Catholique de Louvain CECAM 2008 Developer School : The
More information221A Lecture Notes Variational Method
1 Introduction 1A Lecture Notes Variational Method Most of the problems in physics cannot be solved exactly, and hence need to be dealt with approximately. There are two common methods used in quantum
More informationStructure Factors 59553 78
78 Structure Factors Until now, we have only typically considered reflections arising from planes in a hypothetical lattice containing one atom in the asymmetric unit. In practice we will generally deal
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationThe nearlyfree electron model
Handout 3 The nearlyfree electron model 3.1 Introduction Having derived Bloch s theorem we are now at a stage where we can start introducing the concept of bandstructure. When someone refers to the bandstructure
More informationElectric Dipole moments as probes of physics beyond the Standard Model
Electric Dipole moments as probes of physics beyond the Standard Model K. V. P. Latha NonAccelerator Particle Physics Group Indian Institute of Astrophysics Plan of the Talk Parity (P) and Timereversal
More informationDarrick Chang ICFO The Institute of Photonic Sciences Barcelona, Spain. April 2, 2014
Darrick Chang ICFO The Institute of Photonic Sciences Barcelona, Spain April 2, 2014 ICFO The Institute of Photonic Sciences 10 minute walk 11 years old 22 Research Groups 300 people Research themes: Quantum
More information5.111 Principles of Chemical Science
MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.111 Lecture Summary
More informationHard Condensed Matter WZI
Hard Condensed Matter WZI Tom Gregorkiewicz University of Amsterdam VULaserLab Dec 10, 2015 Hard Condensed Matter Cluster Quantum Matter Optoelectronic Materials Quantum Matter Amsterdam Mark Golden Anne
More informationDFT in practice : Part I. Ersen Mete
plane wave expansion & the Brillouin zone integration Department of Physics Balıkesir University, Balıkesir  Turkey August 13, 2009  NanoDFT 09, İzmir Institute of Technology, İzmir Outline Plane wave
More informationComputer lab: Density functional perturbation theory. theory for lattice dynamics
Computer lab: density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy) Outline 1 The dynamical matrix 2 3 4 5 Dynamical matrix We want to write a small computer
More informationTheoretical approaches to the abinitio study of complex systems
Theoretical approaches to the abinitio study of complex systems Olivia Pulci European Theoretical Spectroscopy Facilty (ETSF), and CNRINFM, Dipartimento di Fisica Università di Roma Tor Vergata NASTISM
More informationElectronic Structure Methods. by Daniel Rohr Vrije Universiteit Amsterdam
Electronic Structure Methods by Daniel Rohr Vrije Universiteit Amsterdam drohr@few.vu.nl References WFT Szaboo & Ostlund Modern Quantum Chemistry Helgaker, Jørgensen & Olsen Molecular ElectronicStructure
More informationLecture 2: Semiconductors: Introduction
Lecture 2: Semiconductors: Introduction Contents 1 Introduction 1 2 Band formation in semiconductors 2 3 Classification of semiconductors 5 4 Electron effective mass 10 1 Introduction Metals have electrical
More informationElectromagnetic Radiation
Activity 17 Electromagnetic Radiation Why? Electromagnetic radiation, which also is called light, is an amazing phenomenon. It carries energy and has characteristics of both particles and waves. We can
More informationSpatially separated excitons in 2D and 1D
Spatially separated excitons in 2D and 1D David Abergel March 10th, 2015 D.S.L. Abergel 3/10/15 1 / 24 Outline 1 Introduction 2 Spatially separated excitons in 2D The role of disorder 3 Spatially separated
More informationThe quantum mechanics of particles in a periodic potential: Bloch s theorem
Handout 2 The quantum mechanics of particles in a periodic potential: Bloch s theorem 2.1 Introduction and health warning We are going to set up the formalism for dealing with a periodic potential; this
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infrared spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More informationCHEM6085: Density Functional Theory Lecture 2. Hamiltonian operators for molecules
CHEM6085: Density Functional Theory Lecture 2 Hamiltonian operators for molecules C.K. Skylaris 1 The (timeindependent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction
More informationBoltzmann Distribution Law
Boltzmann Distribution Law The motion of molecules is extremely chaotic Any individual molecule is colliding with others at an enormous rate Typically at a rate of a billion times per second We introduce
More informationProbing the Vacuum Induced Coherence in a Λsystem
Probing the Vacuum Induced Coherence in a Λsystem Sunish Menon and G. S. Agarwal Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India. (February 8, 1999) We propose a simple test to demonstrate
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationThe Passage of Fast Electrons Through Matter
The Passage of Fast Electrons Through Matter Adam P. Sorini A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2008 Program
More informationChapter 7: Polarization
Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces
More informationSimulation of infrared and Raman spectra
Simulation of infrared and Raman spectra, 1 Bernard Kirtman, 2 Michel Rérat, 3 Simone Salustro, 1 Marco De La Pierre, 1 Roberto Orlando, 1 Roberto Dovesi 1 1) Dipartimento di Chimica, Università di Torino
More informationSolid State Detectors = SemiConductor based Detectors
Solid State Detectors = SemiConductor based Detectors Materials and their properties Energy bands and electronic structure Charge transport and conductivity Boundaries: the pn junction Charge collection
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More informationChemistry 102 Summary June 24 th. Properties of Light
Chemistry 102 Summary June 24 th Properties of Light  Energy travels through space in the form of electromagnetic radiation (EMR).  Examples of types of EMR: radio waves, xrays, microwaves, visible
More informationFirst Principles Calculations of NMR Chemical Shifts. Daniel Sebastiani
First Principles Calculations of NMR Chemical Shifts Methods and Applications Daniel Sebastiani Approche théorique et expérimentale des phénomènes magnétiques et des spectroscopies associées Max Planck
More informationBroadband THz Generation from Photoconductive Antenna
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 2226 331 Broadband THz Generation from Photoconductive Antenna Qing Chang 1, Dongxiao Yang 1,2, and Liang Wang 1 1 Zhejiang
More informationThe plasmoelectric effect: optically induced electrochemical potentials in resonant metallic structures
The plasmoelectric effect: optically induced electrochemical potentials in resonant metallic structures Matthew T. Sheldon and Harry A. Atwater Thomas J. Watson Laboratories of Applied Physics, California
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More information2. Molecular stucture/basic
2. Molecular stucture/basic spectroscopy The electromagnetic spectrum Spectral region for atomic and molecular spectroscopy E. Hecht (2nd Ed.) Optics, AddisonWesley Publishing Company,1987 Spectral regions
More informationChapter 8 Molecules. Some molecular bonds involve sharing of electrons between atoms. These are covalent bonds.
Chapter 8 Molecules (We have only three days for chapter 8!) 8.1 The Molecular Bond A molecule is an electrically neutral group of atoms held together strongly enough to behave as a single particle. A
More informationSpecific Intensity. I ν =
Specific Intensity Initial question: A number of active galactic nuclei display jets, that is, long, nearly linear, structures that can extend for hundreds of kiloparsecs. Many have two oppositelydirected
More informationL5. P1. Lecture 5. Solids. The free electron gas
Lecture 5 Page 1 Lecture 5 L5. P1 Solids The free electron gas In a solid state, a few loosely bound valence (outermost and not in completely filled shells) elections become detached from atoms and move
More informationF en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.
The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should
More informationIndiana's Academic Standards 2010 ICP Indiana's Academic Standards 2016 ICP. map) that describe the relationship acceleration, velocity and distance.
.1.1 Measure the motion of objects to understand.1.1 Develop graphical, the relationships among distance, velocity and mathematical, and pictorial acceleration. Develop deeper understanding through representations
More informationLaws of Thermodynamics
Laws of Thermodynamics Thermodynamics Thermodynamics is the study of the effects of work, heat, and energy on a system Thermodynamics is only concerned with macroscopic (largescale) changes and observations
More informationarxiv:1204.4608v1 [condmat.mtrlsci] 20 Apr 2012
arxiv:1204.4608v1 [condmat.mtrlsci] 20 Apr 2012 GW quasiparticle band gaps of anatase TiO 2 starting from DFT+U Christopher E. Patrick and Feliciano Giustino Department of Materials, University of Oxford,
More informationNumerical Analysis of Perforated Microring Resonator Based Refractive Index Sensor
Numerical Analysis of Perforated Microring Resonator Based Refractive Index Sensor M. Gabalis *1, D. Urbonas 1, and R. Petruškevičius 1 1 Institute of Physics of Center for Physical Sciences and Technology,
More informationphys4.17 Page 1  under normal conditions (pressure, temperature) graphite is the stable phase of crystalline carbon
Covalent Crystals  covalent bonding by shared electrons in common orbitals (as in molecules)  covalent bonds lead to the strongest bound crystals, e.g. diamond in the tetrahedral structure determined
More informationAn Introduction to HartreeFock Molecular Orbital Theory
An Introduction to HartreeFock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction HartreeFock theory is fundamental
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More informationThe Schwinger Mechanism and Graphene. D. Allor *, T.DC, D. A. McGady * arxiv:0708.1471 * University of Maryland Undergrads
The Schwinger Mechanism and Graphene & D. Allor *, T.DC, D. A. McGady * arxiv:0708.1471 * University of Maryland Undergrads Outline What is the Schwinger Mechanism? Why is it worth worrying about? How
More informationChemical Synthesis. Overview. Chemical Synthesis of Nanocrystals. SelfAssembly of Nanocrystals. Example: Cu 146 Se 73 (PPh 3 ) 30
Chemical Synthesis Spontaneous organization of molecules into stable, structurally welldefined aggregates at the nanometer length scale. Overview The 1100 nm nanoscale length is in between traditional
More informationChapter 18 Electric Forces and Electric Fields. Key Concepts:
Chapter 18 Lectures Monday, January 25, 2010 7:33 AM Chapter 18 Electric Forces and Electric Fields Key Concepts: electric charge principle of conservation of charge charge polarization, both permanent
More informationNMR SPECTROSCOPY. Basic Principles, Concepts, and Applications in Chemistry. Harald Günther University of Siegen, Siegen, Germany.
NMR SPECTROSCOPY Basic Principles, Concepts, and Applications in Chemistry Harald Günther University of Siegen, Siegen, Germany Second Edition Translated by Harald Günther JOHN WILEY & SONS Chichester
More information4. The Infinite Square Well
4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the
More information thus, the total number of atoms per second that absorb a photon is
Stimulated Emission of Radiation  stimulated emission is referring to the emission of radiation (a photon) from one quantum system at its transition frequency induced by the presence of other photons
More informationENERGY TRANSFER IN THE WEAK AND STRONG COUPLING REGIME
ERC Starting Grant 2011 Dipar)mento di Scienze Chimiche Università degli Studi di Padova via Marzolo 1, 35131 Padova Italy ENERGY TRANSFER IN THE WEAK AND STRONG COUPLING REGIME [1] Vekshin, N. L. Energy
More informationSolid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16
Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 MaxPlanck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm. 02.001 Luruper Chaussee 149
More informationLecture 1: Microscopic Theory of Radiation
253a: QFT Fall 2009 Matthew Schwartz Lecture : Microscopic Theory of Radiation Blackbody Radiation Quantum Mechanics began on October 9, 900 with Max Planck s explanation of the blackbody radiation spectrum.
More informationAtomic Structure: Chapter Problems
Atomic Structure: Chapter Problems Bohr Model Class Work 1. Describe the nuclear model of the atom. 2. Explain the problems with the nuclear model of the atom. 3. According to Niels Bohr, what does n stand
More informationSymmetric Stretch: allows molecule to move through space
BACKGROUND INFORMATION Infrared Spectroscopy Before introducing the subject of IR spectroscopy, we must first review some aspects of the electromagnetic spectrum. The electromagnetic spectrum is composed
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More information3.003 Lab 4 Simulation of Solar Cells
Mar. 9, 2010 Due Mar. 29, 2010 3.003 Lab 4 Simulation of Solar Cells Objective: To design a silicon solar cell by simulation. The design parameters to be varied in this lab are doping levels of the substrate
More informationEnergy. Mechanical Energy
Principles of Imaging Science I (RAD119) Electromagnetic Radiation Energy Definition of energy Ability to do work Physicist s definition of work Work = force x distance Force acting upon object over distance
More informationPhotoinduced volume change in chalcogenide glasses
Photoinduced volume change in chalcogenide glasses (Ph.D. thesis points) Rozália Lukács Budapest University of Technology and Economics Department of Theoretical Physics Supervisor: Dr. Sándor Kugler 2010
More informationThe Physics of Energy sources Renewable sources of energy. Solar Energy
The Physics of Energy sources Renewable sources of energy Solar Energy B. Maffei Bruno.maffei@manchester.ac.uk Renewable sources 1 Solar power! There are basically two ways of using directly the radiative
More informationVIII.4. Field Effect Transistors
Field Effect Transistors (FETs) utilize a conductive channel whose resistance is controlled by an applied potential. 1. Junction Field Effect Transistor (JFET) In JFETs a conducting channel is formed of
More informationSpectroscopic Ellipsometry:
Spectroscopic : What it is, what it will do, and what it won t do by Harland G. Tompkins Introduction Fundamentals Anatomy of an ellipsometric spectrum Analysis of an ellipsometric spectrum What you can
More informationCollaborative software development for nanoscale physics
Collaborative software development for nanoscale physics Yann Pouillon mailto:yann.pouillon@ehu.es NanoBio Spectroscopy Group Universidad del País Vasco (UPV/EHU) & CSIC, DonostiaSan Sebastián, Spain
More informationIntroduction to Quantum Dot Nanocrystals and Nanocrystal Solids. Nuri Yazdani, 10.03.15
Introduction to Quantum Dot Nanocrystals and Nanocrystal Solids Nuri Yazdani, 10.03.15 What is a QD Nanocrystal Time: ~15m What is a QD nanocrystal? Bulk Crystal Periodic lattice of atoms which extends
More informationThe Role of Electric Polarization in Nonlinear optics
The Role of Electric Polarization in Nonlinear optics Sumith Doluweera Department of Physics University of Cincinnati Cincinnati, Ohio 45221 Abstract Nonlinear optics became a very active field of research
More informationPhysics 551: Solid State Physics F. J. Himpsel
Physics 551: Solid State Physics F. J. Himpsel Background Most of the objects around us are in the solid state. Today s technology relies heavily on new materials, electronics is predominantly solid state.
More informationApplications of Quantum Chemistry HΨ = EΨ
Applications of Quantum Chemistry HΨ = EΨ Areas of Application Explaining observed phenomena (e.g., spectroscopy) Simulation and modeling: make predictions New techniques/devices use special quantum properties
More informationThe First Law of Thermodynamics: Closed Systems. Heat Transfer
The First Law of Thermodynamics: Closed Systems The first law of thermodynamics can be simply stated as follows: during an interaction between a system and its surroundings, the amount of energy gained
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for nonphysics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationApplied Physics of solar energy conversion
Applied Physics of solar energy conversion Conventional solar cells, and how lazy thinking can slow you down Some new ideas *************************************************************** Our work on semiconductor
More informationElectronic Structure and Transition Intensities in RareEarth Materials
Electronic Structure and Transition Intensities in RareEarth Materials Michael F Reid Department of Physics and Astronomy and MacDiarmid Institute for Advanced Materials and Nanotechnology University
More informationFundamentals of molecular absorption spectroscopy (UV/VIS)
10.2.1.3 Molecular spectroscopy 10.2.1.3.1 Introduction Molecular radiation results from the rotational, vibrational and electronic energy transitions of molecules. Band spectra are the combination of
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More information particle with kinetic energy E strikes a barrier with height U 0 > E and width L.  classically the particle cannot overcome the barrier
Tunnel Effect:  particle with kinetic energy E strikes a barrier with height U 0 > E and width L  classically the particle cannot overcome the barrier  quantum mechanically the particle can penetrated
More informationChapter 26. Capacitance and Dielectrics
Chapter 26 Capacitance and Dielectrics Capacitors Capacitors are devices that store electric charge Examples where capacitors are used: radio receivers filters in power supplies energystoring devices
More informationLecture 1: Introduction to Random Walks and Diffusion
Lecture : Introduction to Random Walks and Diffusion Scribe: Chris H. Rycroft (and Martin Z. Bazant) Department of Mathematics, MIT February, 5 History The term random walk was originally proposed by Karl
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationTobias Märkl. November 16, 2009
,, Tobias Märkl to 1/f November 16, 2009 1 / 33 Content 1 duction to of Statistical Comparison to Other Types of Noise of of 2 Random duction to Random General of, to 1/f 3 4 2 / 33 , to 1/f 3 / 33 What
More informationBlackbody radiation derivation of Planck s radiation low
Blackbody radiation derivation of Planck s radiation low 1 Classical theories of Lorentz and Debye: Lorentz (oscillator model): Electrons and ions of matter were treated as a simple harmonic oscillators
More informationRadiation Interactions with Matter: Energy Deposition
Radiation Interactions with Matter: Energy Deposition Biological effects are the end product of a long series of phenomena, set in motion by the passage of radiation through the medium. Image removed due
More information5.6 Binary rare earth alloys
5.6 BINARY RARE EARTH ALLOYS 247 5.6 Binary rare earth alloys The great similarity in the chemical properties of the different rare earth metals allows almost complete mutual solubility. It is therefore
More informationAb Initio SecondOrder Nonlinear Optics in Solids: SecondHarmonic Generation Spectroscopy from TimeDependent DensityFunctional Theory.
Ab Initio SecondOrder Nonlinear Optics in Solids: SecondHarmonic Generation Spectroscopy from TimeDependent DensityFunctional Theory Eleonora Luppi, Hannes Hübener, and Valérie Véniard Laboratoire
More informationOrbits of the LennardJones Potential
Orbits of the LennardJones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The LennardJones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials
More informationElectrical Conductivity
Advanced Materials Science  Lab Intermediate Physics University of Ulm Solid State Physics Department Electrical Conductivity Translated by MichaelStefan Rill January 20, 2003 CONTENTS 1 Contents 1 Introduction
More information