ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles Trieste, 19-23 March 2018
Lecture Thu.1 Electron-phonon effects in ARPES and IXS Feliciano Giustino Department of Materials, University of Oxford Department of Materials Science and Engineering, Cornell University Giustino, Lecture Thu.1 02/36
Lecture Summary • Satellites in photoelectron spectroscopy • Phonon Green’s function and self-energy • Connection with density-functional perturbation theory • Non-adiabatic phonons • Phonon lifetimes • Electron-phonon matrix element and Fr¨ ohlich interaction Giustino, Lecture Thu.1 03/36
Angle-resolved photoelectron spectroscopy (ARPES) commons.wikimedia.org/wiki/File:ARPESgeneral.png Giustino, Lecture Thu.1 04/36
ARPES kinks and satellites � ω ph ≪ ε F : Kinks Figure from Giustino, Rev. Mod. Phys. 89, 015003 (2017) Giustino, Lecture Thu.1 05/36
ARPES kinks and satellites � ω ph ≪ ε F : Kinks � ω ph ∼ ε F : Satellites Figure from Giustino, Rev. Mod. Phys. 89, 015003 (2017) Giustino, Lecture Thu.1 05/36
ARPES on doped transition-metal oxides Conduction band Fermi level Energy Momentum Giustino, Lecture Thu.1 06/36
ARPES on doped transition-metal oxides Conduction band Fermi level Energy Satellite Satellite Momentum Giustino, Lecture Thu.1 06/36
ARPES on doped transition-metal oxides • Example: SrTiO 3 (001) surface Figure from Wang et al, Nature Mater. 15, 835 (2016) Giustino, Lecture Thu.1 07/36
Diagrammatic representation of the self-energy Standard GW self-energy (we will ignore this from now on) Fan-Migdal self-energy Debye-Waller self-energy Giustino, Lecture Thu.1 08/36
Cumulant expansion method Aryasetiawan et al, Phys. Rev. Lett. 77, 2268 (1996); Gumhalter et al, Phys. Rev. B 94, 035103 (2016); Zhou et al, J. Chem. Phys. 143, 184109 (2015); Nery et al, arXiv:1710.07594 (2017); & Giustino, Lecture Thu.1 09/36
Cumulant expansion method Σ FM n k ( ω ) Aryasetiawan et al, Phys. Rev. Lett. 77, 2268 (1996); Gumhalter et al, Phys. Rev. B 94, 035103 (2016); Zhou et al, J. Chem. Phys. 143, 184109 (2015); Nery et al, arXiv:1710.07594 (2017); & Giustino, Lecture Thu.1 09/36
Cumulant expansion method Cumulant Σ FM n k ( ω ) formula Aryasetiawan et al, Phys. Rev. Lett. 77, 2268 (1996); Gumhalter et al, Phys. Rev. B 94, 035103 (2016); Zhou et al, J. Chem. Phys. 143, 184109 (2015); Nery et al, arXiv:1710.07594 (2017); & Giustino, Lecture Thu.1 09/36
Cumulant expansion method Cumulant Σ FM n k ( ω ) A k ( ω ) with satellites formula Aryasetiawan et al, Phys. Rev. Lett. 77, 2268 (1996); Gumhalter et al, Phys. Rev. B 94, 035103 (2016); Zhou et al, J. Chem. Phys. 143, 184109 (2015); Nery et al, arXiv:1710.07594 (2017); & Giustino, Lecture Thu.1 09/36
Cumulant expansion method • Example: n-doped TiO 2 anatase ARPES experiment Calculation Moser et al, PRL 110, 196403 (2013) Figure from Verdi et al, Nat. Commun. 8, 15769 (2017) Giustino, Lecture Thu.1 10/36
Lattice Vibrations Electrons Giustino, Lecture Thu.1 11/36
Lattice Vibrations Electrons Giustino, Lecture Thu.1 11/36
Time-evolution of atomic displacements Central quantity to study phonons in a many-body framework: the displacement-displacement correlation function (Lecture Wed.1) D κκ ′ ( tt ′ ) = − i � � ˆ κ ′ ( t ′ ) � τ T T ∆ˆ τ κ ( t ) ∆ˆ 3 × 3 matrices in the Cartesian coordinates Giustino, Lecture Thu.1 12/36
Time-evolution of atomic displacements Central quantity to study phonons in a many-body framework: the displacement-displacement correlation function (Lecture Wed.1) D κκ ′ ( tt ′ ) = − i � � ˆ κ ′ ( t ′ ) � τ T T ∆ˆ τ κ ( t ) ∆ˆ 3 × 3 matrices in the Cartesian coordinates Heisenberg time evolution of atomic displacements i � d τ κ ( t ) , ˆ dt ∆ˆ τ κ ( t ) = [∆ˆ H ] Giustino, Lecture Thu.1 12/36
Time-evolution of atomic displacements Central quantity to study phonons in a many-body framework: the displacement-displacement correlation function (Lecture Wed.1) D κκ ′ ( tt ′ ) = − i � � ˆ κ ′ ( t ′ ) � τ T T ∆ˆ τ κ ( t ) ∆ˆ 3 × 3 matrices in the Cartesian coordinates Heisenberg time evolution of atomic displacements i � d τ κ ( t ) , ˆ dt ∆ˆ τ κ ( t ) = [∆ˆ H ] Make it look like Newton’s equation by taking the 2nd derivative d 2 τ κ = − M κ τ κ , ˆ H ] , ˆ dt 2 ∆ˆ � 2 [[∆ˆ M κ H ] Giustino, Lecture Thu.1 12/36
Time-evolution of atomic displacements Central quantity to study phonons in a many-body framework: the displacement-displacement correlation function (Lecture Wed.1) D κκ ′ ( tt ′ ) = − i � � ˆ κ ′ ( t ′ ) � τ T T ∆ˆ τ κ ( t ) ∆ˆ 3 × 3 matrices in the Cartesian coordinates Heisenberg time evolution of atomic displacements i � d τ κ ( t ) , ˆ dt ∆ˆ τ κ ( t ) = [∆ˆ H ] Make it look like Newton’s equation by taking the 2nd derivative d 2 τ κ = − M κ τ κ , ˆ H ] , ˆ dt 2 ∆ˆ � 2 [[∆ˆ M κ H ] � �� � dimensions of force Giustino, Lecture Thu.1 12/36
Many-body phonon self-energy ∂ 2 ∂t 2 D κκ ′ ( tt ′ ) = M κ Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ � �� � phonon self-energy Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ � �� � phonon self-energy � ∂ 2 e 2 Z κ Z κ ′ d r ǫ − 1 Π κκ ′ ( ω ) = e ( τ κ , r , ω ) ∂ τ κ ∂ τ T 4 πǫ 0 | r − τ κ ′ | κ ′ Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ � �� � phonon self-energy � ∂ 2 e 2 Z κ Z κ ′ d r ǫ − 1 Π κκ ′ ( ω ) = e ( τ κ , r , ω ) 4 πǫ 0 | r − τ κ ′ | − (self force) ∂ τ κ ∂ τ T κ ′ Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ � �� � phonon self-energy � ∂ 2 e 2 Z κ Z κ ′ d r ǫ − 1 Π κκ ′ ( ω ) = e ( τ κ , r , ω ) 4 πǫ 0 | r − τ κ ′ | − (self force) ∂ τ κ ∂ τ T κ ′ Π contains the spring constants for a Coulomb interaction between nuclei, screened by the electronic dielectric matrix Giustino, Lecture Thu.1 13/36
Many-body phonon self-energy Using Schwinger’s functional derivative technique � ∂ 2 � dt ′′ Π κκ ′′ ( tt ′′ ) D κ ′′ κ ′ ( t ′′ t ′ ) ∂t 2 D κκ ′ ( tt ′ ) = − I 3 × 3 δ κκ ′ δ ( tt ′ ) − M κ κ ′′ � �� � phonon self-energy � ∂ 2 e 2 Z κ Z κ ′ d r ǫ − 1 Π κκ ′ ( ω ) = e ( τ κ , r , ω ) 4 πǫ 0 | r − τ κ ′ | − (self force) ∂ τ κ ∂ τ T κ ′ Π contains the spring constants for a Coulomb interaction between nuclei, screened by the electronic dielectric matrix → ∂ 2 E tot example: E tot = 1 2 C ( τ − τ 0 ) 2 − − = C ∂τ 2 Giustino, Lecture Thu.1 13/36
Many-body vibrational eigenfrequencies D 11 D 12 . . . D 1 N . . . D 21 D 22 D 2 N D = . . . ... . . . . . . D N 1 D N 2 . . . D NN 3 N × 3 N Giustino, Lecture Thu.1 14/36
Many-body vibrational eigenfrequencies D 11 D 12 . . . D 1 N Π 11 Π 12 . . . Π 1 N . . . . . . D 21 D 22 D 2 N Π 21 Π 22 Π 2 N D = Π = . . . . . . ... ... . . . . . . . . . . . . D N 1 D N 2 . . . D NN Π N 1 Π N 2 . . . Π NN 3 N × 3 N 3 N × 3 N Giustino, Lecture Thu.1 14/36
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