Introduction BSE Response In practice Bethe-Salpeter equation: electron-hole excitations and optical spectra Ilya Tokatly European Theoretical Spectroscopy Facility (ETSF) NanoBio Spectroscopy Group - UPV/EHU San Sebastiàn - Spain IKERBASQUE, Basque Foundation for Science - Bilbao - Spain ilya.tokatly@ehu.es TDDFT school - Benasque 2014
bg=white Introduction BSE Response In practice Outline Optics and two-particle dynamics: Why BSE? 1 The Bethe-Salpeter equation: Pictorial derivation 2 Macroscopic response and the Bethe-Salpeter equation 3 The Bethe-Salpeter equation in practice 4
bg=white Introduction BSE Response In practice Outline Optics and two-particle dynamics: Why BSE? 1 The Bethe-Salpeter equation: Pictorial derivation 2 Macroscopic response and the Bethe-Salpeter equation 3 The Bethe-Salpeter equation in practice 4
bg=white Introduction BSE Response In practice Historical remark Original Bethe-Salpeter equation In 1951 [Phys. Rev. 84 , 1232 (1951)] Bethe and Salpeter derived an equation describing propagation of two interacting relativistic particles. The physical motivation was the problem of deuteron – a bound state of two neucleons (proton and neutron in the nucleus of deuterium.) Why this equation is so important in the theory of optical spectra?
bg=white Introduction BSE Response In practice Historical remark Original Bethe-Salpeter equation In 1951 [Phys. Rev. 84 , 1232 (1951)] Bethe and Salpeter derived an equation describing propagation of two interacting relativistic particles. The physical motivation was the problem of deuteron – a bound state of two neucleons (proton and neutron in the nucleus of deuterium.) Why this equation is so important in the theory of optical spectra?
bg=white Introduction BSE Response In practice Optical absorption: Experiment and Phenomenology Light is absorbed: I = I 0 e − α ( ω ) x Classical electrodynamics q 2 = ω 2 E = E 0 e − i ( ωt − qx ) , c 2 ǫ M ( ω ) ǫ M ( ω ) = ǫ ′ M ( ω ) + iǫ ′′ M ( ω ) q ≈ ω � ω 2 c √ ǫ ′ M + i M ǫ ′′ M c ǫ ′ � ǫ ′ M = n r – index of refraction I ∼ | E | 2 = | E 0 | 2 e − α ( ω ) x ω α ( ω ) = cn r ǫ ′′ M ( ω ) ǫ ′′ M ( ω ) ∼ absorption rate Exp. at 30 K from: P . Lautenschlager et al. , Phys. Rev. B 36 , 4821 (1987).
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair Representation by Feynman diagrams: photon creates an e-h pair the pair propagates freely it recombines and recreates a photon
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair Representation by Feynman diagrams: photon creates an e-h pair the pair propagates freely it recombines and recreates a photon
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair Representation by Feynman diagrams: photon creates an e-h pair the pair propagates freely it recombines and recreates a photon
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair Representation by Feynman diagrams: photon creates an e-h pair the pair propagates freely it recombines and recreates a photon
bg=white Introduction BSE Response In practice Optical absorption: Microscopic picture Elementary process of absorption: Photon creates a single e-h pair Representation by Feynman diagrams: photon creates an e-h pair the pair propagates freely it recombines and recreates a photon Absorption rate is given by an imaginary part of the polarization loop W = 2 π � v | ϕ j �| 2 δ ( ε j − ε i − � ω ) ∼ Im ǫ ( ω ) |� ϕ i | e · ˆ � i,j
bg=white Introduction BSE Response In practice Absorption by independent Kohn-Sham particles Independent transitions: ǫ ′′ ( ω ) = 8 π 2 � v | ϕ i �| 2 δ ( ε j − ε i − ω ) |� ϕ j | e · ˆ ω 2 ij
bg=white Introduction BSE Response In practice Absorption by independent Kohn-Sham particles Independent transitions: ǫ ′′ ( ω ) = 8 π 2 � v | ϕ i �| 2 δ ( ε j − ε i − ω ) |� ϕ j | e · ˆ ω 2 ij Particles are interacting!
bg=white Introduction BSE Response In practice Interaction effects: self-energy corrections 1st class of interaction corrections: Created electron and hole interact with other particles in the system, but do not touch each other
bg=white Introduction BSE Response In practice Interaction effects: self-energy corrections 1st class of interaction corrections: Created electron and hole interact with other particles in the system, but do not touch each other Absorption by “dressed” particles = + Bare propagator G 0 is replaced by the full propagator G = G 0 + G 0 Σ G � [ ω − ˆ h 0 ( r )] G ( r , r ′ , ω ) + d r 1 Σ( r , r 1 , ω ) G ( r 1 , r ′ , ω ) = δ ( r − r ′ )
bg=white Introduction BSE Response In practice Self-energy corrections Perturbative GW corrections ˆ h 0 ( r ) ϕ i ( r ) + V xc ( r ) ϕ i ( r ) = ǫ i ϕ i ( r ) � d r ′ Σ( r , r ′ , ω = E i ) φ i ( r ′ ) ˆ h 0 ( r ) φ i ( r ) + = E i φ i ( r ) First-order perturbative corrections with Σ = GW : E i − ǫ i = � ϕ i | Σ − V xc | ϕ i � Hybersten and Louie, PRB 34 (1986); Godby, Schlüter and Sham, PRB 37 (1988)
bg=white Introduction BSE Response In practice Optical absorption: Independent quasiparticles Independent transitions: ǫ ′′ ( ω ) = 8 π 2 � v | ϕ i �| 2 δ ( E j − E i − ω ) |� ϕ j | e · ˆ ω 2 ij
bg=white Introduction BSE Response In practice Interaction effects: vertex (excitonic) corrections 2nd class of interaction corrections: includes all direct and indirect interactions between electron and hole created by a photon
bg=white Introduction BSE Response In practice Interaction effects: vertex (excitonic) corrections 2nd class of interaction corrections: includes all direct and indirect interactions between electron and hole created by a photon Summing up all such interaction processes we get: Empty polarization loop is replaced by the full two-particle propagator L ( r 1 t 1 ; r 2 t 2 ; r 3 t 3 ; r 4 t 4 ) = L (1234) with joined ends
bg=white Introduction BSE Response In practice Interaction effects: vertex (excitonic) corrections 2nd class of interaction corrections: includes all direct and indirect interactions between electron and hole created by a photon Summing up all such interaction processes we get: Empty polarization loop is replaced by the full two-particle propagator L ( r 1 t 1 ; r 2 t 2 ; r 3 t 3 ; r 4 t 4 ) = L (1234) with joined ends Equation for L(1234) is the Bethe-Salpeter equation!
bg=white Introduction BSE Response In practice Absorption Neutral excitations → poles of two-particle Green’s function L Excitonic effects = electron - hole interaction
bg=white Introduction BSE Response In practice Absorption Neutral excitations → poles of two-particle Green’s function L Excitonic effects = electron - hole interaction
bg=white Introduction BSE Response In practice Absorption Neutral excitations → poles of two-particle Green’s function L Excitonic effects = electron - hole interaction
bg=white Introduction BSE Response In practice Outline Optics and two-particle dynamics: Why BSE? 1 The Bethe-Salpeter equation: Pictorial derivation 2 Macroscopic response and the Bethe-Salpeter equation 3 The Bethe-Salpeter equation in practice 4
bg=white Introduction BSE Response In practice Derivation of the Bethe-Salpeter equation (1) Propagator of e-h pair in a many-body system: Solid lines stand for bare one-particle Green’s functions G 0 (12) = G 0 ( r 1 , r 2 , t 1 − t 2 ) Wiggled lines correspond to the interaction (Coulomb) potential e 2 v (12) = v ( r 1 − r 2 ) δ ( t 1 − t 2 ) = | r 1 − r 2 | δ ( t 1 − t 2 ) Integration over space-time coordinates of all intermediate points in each graph is assumed
bg=white Introduction BSE Response In practice Derivation of the Bethe-Salpeter equation (1) Propagator of e-h pair in a many-body system: 1st step: Dressing one-particle propagators Self-energy Σ is a sum of all 1-particle irreducible diagrams Full 1-particle Green’s function satisfies the Dyson equation = +
bg=white Introduction BSE Response In practice Derivation of the Bethe-Salpeter equation (2) Propagation of dressed interacting electron and hole: 2nd step: Classification of scattering processes At this stage we identify two-particle irreducible blocks where γ (1234) of the electron-hole stattering amplitude
bg=white Introduction BSE Response In practice Derivation of the Bethe-Salpeter equation (3) Final step: Summation of a geometric series The result is the Bethe-Salpeter equation
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