ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles Trieste, 19-23 March 2018
Lecture Tue.1 Introduction to electron-phonon interactions Feliciano Giustino Department of Materials, University of Oxford Department of Materials Science and Engineering, Cornell University Giustino, Lecture Tue.1 02/31
Lecture Summary • Manifestations of the electron-phonon interaction • Rayleigh-Schr¨ odinger perturbation theory • The electron-phonon matrix element • Brillouin-zone integrals and Wannier interpolation • The electron-phonon coupling constant • Connection with molecular dynamics simulations Giustino, Lecture Tue.1 03/31
Where do electron-phonon interactions come from? Giustino, Lecture Tue.1 04/31
Ionic degrees of freedom in the Kohn-Sham equations − � 2 ∇ 2 ψ n + V SCF ψ n = E n ψ n 2 m e Giustino, Lecture Tue.1 05/31
Ionic degrees of freedom in the Kohn-Sham equations − � 2 ∇ 2 ψ n + V SCF ψ n = E n ψ n 2 m e � | ψ n ( r ) | 2 n ( r ) = n ∈ occ Giustino, Lecture Tue.1 05/31
Ionic degrees of freedom in the Kohn-Sham equations − � 2 ∇ 2 ψ n + V SCF ψ n = E n ψ n 2 m e � | ψ n ( r ) | 2 n ( r ) = n ∈ occ � n ( r ′ ) d r ′ V SCF ( r ) = − e 2 � � � Z κ | r − τ κ | − + V xc [ n ( r )] | r − r ′ | 4 πǫ 0 κ Giustino, Lecture Tue.1 05/31
Ionic degrees of freedom in the Kohn-Sham equations − � 2 ∇ 2 ψ n + V SCF ψ n = E n ψ n 2 m e � | ψ n ( r ) | 2 n ( r ) = n ∈ occ � n ( r ′ ) d r ′ V SCF ( r ) = − e 2 � � � Z κ | r − τ κ | − + V xc [ n ( r )] | r − r ′ | 4 πǫ 0 κ Giustino, Lecture Tue.1 05/31
Ionic degrees of freedom in the Kohn-Sham equations The SCF potential depends parametrically on the ionic coordinates V SCF ( r ; τ 1 , τ 2 , · · · ) Giustino, Lecture Tue.1 06/31
Ionic degrees of freedom in the Kohn-Sham equations The SCF potential depends parametrically on the ionic coordinates V SCF ( r ; τ 1 , τ 2 , · · · ) • Consider only one ion and one Cartesian direction for simplicity • Displace atoms from equilibrium sites, τ = τ 0 + u Giustino, Lecture Tue.1 06/31
Ionic degrees of freedom in the Kohn-Sham equations The SCF potential depends parametrically on the ionic coordinates V SCF ( r ; τ 1 , τ 2 , · · · ) • Consider only one ion and one Cartesian direction for simplicity • Displace atoms from equilibrium sites, τ = τ 0 + u ∂ 2 V SCF V SCF ( r ; τ ) = V SCF ( r ; τ 0 ) + ∂V SCF u + 1 u 2 + · · · ∂τ 2 ∂τ 2 Giustino, Lecture Tue.1 06/31
Ionic degrees of freedom in the Kohn-Sham equations The SCF potential depends parametrically on the ionic coordinates V SCF ( r ; τ 1 , τ 2 , · · · ) • Consider only one ion and one Cartesian direction for simplicity • Displace atoms from equilibrium sites, τ = τ 0 + u ∂ 2 V SCF V SCF ( r ; τ ) = V SCF ( r ; τ 0 ) + ∂V SCF u + 1 u 2 + · · · ∂τ 2 ∂τ 2 Perturbation Hamiltonian leading to EPIs Giustino, Lecture Tue.1 06/31
Some manifestations of electron-phonon interactions • Electron mobility in monolayer and bilayer MoS 2 Figure from Baugher et al, Nano Lett. 13, 4212 (2013) Giustino, Lecture Tue.1 07/31
Some manifestations of electron-phonon interactions • Phonon-assisted optical absorption in silicon Data from Green et al, Prog. Photovolt. Res. Appl. 3, 189 (1995) Giustino, Lecture Tue.1 08/31
Some manifestations of electron-phonon interactions • High-temperature superconductivity in compressed H 3 S Figure from Drozdov et al, Nature 73, 525 (2015) Giustino, Lecture Tue.1 09/31
Some manifestations of electron-phonon interactions • Temperature-dependent photoluminescence in hybrid perovskites Figure from Wright et al, Nat. Commun. 7, 11755 (2016) Giustino, Lecture Tue.1 10/31
Some manifestations of electron-phonon interactions • Electron mass enhancement in MgB 2 Figure from Mou et al, Phys. Rev. B 91, 140502(R) (2015) Giustino, Lecture Tue.1 11/31
Rayleigh-Schr¨ odinger perturbation theory ∂ 2 V SCF H ep = ∂V SCF u + 1 u 2 + · · · ∆ ˆ 2 ∂τ 2 ∂τ Giustino, Lecture Tue.1 12/31
Rayleigh-Schr¨ odinger perturbation theory ∂ 2 V SCF H ep = ∂V SCF u + 1 u 2 + · · · ∆ ˆ 2 ∂τ 2 ∂τ ∆ E n = � n | ∂V SCF • Energies u | n � ∂τ Giustino, Lecture Tue.1 12/31
Rayleigh-Schr¨ odinger perturbation theory ∂ 2 V SCF H ep = ∂V SCF u + 1 u 2 + · · · ∆ ˆ 2 ∂τ 2 ∂τ ∆ E n = � n | ∂V SCF • Energies u | n � ∂τ � m | ∂V SCF u | n � ∂τ � • Wavefunctions ∆ ψ n ( r ) = ψ m ( r ) E n − E m m � = n Giustino, Lecture Tue.1 12/31
Rayleigh-Schr¨ odinger perturbation theory ∂ 2 V SCF H ep = ∂V SCF u + 1 u 2 + · · · ∆ ˆ 2 ∂τ 2 ∂τ ∆ E n = � n | ∂V SCF • Energies u | n � ∂τ � m | ∂V SCF u | n � ∂τ � • Wavefunctions ∆ ψ n ( r ) = ψ m ( r ) E n − E m m � = n Γ n → m = 2 π � |� m | ∂V SCF u | n �| 2 δ ( E m − E n − � ω ) • Transition rates ∂τ Giustino, Lecture Tue.1 12/31
Thermodynamic averages What is the atomic displacement u in ∆ ˆ H ep ? Giustino, Lecture Tue.1 13/31
Thermodynamic averages What is the atomic displacement u in ∆ ˆ H ep ? C = Mω 2 M u Giustino, Lecture Tue.1 13/31
Thermodynamic averages What is the atomic displacement u in ∆ ˆ H ep ? C = Mω 2 M u � u 2 � T = k B T classical Mω 2 Giustino, Lecture Tue.1 13/31
Thermodynamic averages What is the atomic displacement u in ∆ ˆ H ep ? C = Mω 2 M u � u 2 � T = k B T quantum classical classical Mω 2 � � ω � � � � � u 2 � T = 2 n + 1 2 Mω k B T Giustino, Lecture Tue.1 13/31
Thermodynamic averages � ∆ E n � T − − − − − − − − − → Temperature-dependent band structures �· · · ∆ ψ n ( r ) · · ·� T − − → Phonon-assisted optical absorption � Γ n → m � T − − − − − − − − − → Phonon-limited carrier mobilities Giustino, Lecture Tue.1 14/31
Temperature-dependent band structures ∆ E n = � n | ∂V SCF | n � u ∂τ Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures ∆ E n = � n | ∂V SCF | n � u ∂τ Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures 2 � � � � m | ∂V SCF � � | n � � � ∆ E n = � n | ∂V SCF ∂τ � � u 2 | n � u + ∂τ E n − E m m � = n Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures 2 � � � � m | ∂V SCF � � | n � � � 2 � n | ∂ 2 V SCF ∆ E n = � n | ∂V SCF ∂τ u 2 +1 � � | n � u 2 | n � u + ∂τ E n − E m ∂τ 2 m � = n Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures 2 � � � � m | ∂V SCF � � | n � � � 2 � n | ∂ 2 V SCF ∆ E n = � n | ∂V SCF ∂τ u 2 +1 � � | n � u 2 | n � u + ∂τ E n − E m ∂τ 2 m � = n 2 � � � � m | ∂V SCF � � | n � � � 2 � n | ∂ 2 V SCF + 1 ∂τ � � � u 2 � T � ∆ E n � T = | n � E n − E m ∂τ 2 m � = n ( Lecture Thu.2) Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures 2 � � � � m | ∂V SCF � � | n � � � 2 � n | ∂ 2 V SCF ∆ E n = � n | ∂V SCF ∂τ u 2 +1 � � | n � u 2 | n � u + ∂τ E n − E m ∂τ 2 m � = n 2 � � � � m | ∂V SCF � � | n � � � 2 � n | ∂ 2 V SCF + 1 ∂τ � � � � ∆ E n � T = | n � 2 Mω (2 n T + 1) E n − E m ∂τ 2 m � = n ( Lecture Thu.2) Giustino, Lecture Tue.1 15/31
Temperature-dependent band structures |· · · | 2 � < 0 E c − E m m � = c |· · · | 2 � > 0 E v − E m m � = v ( Lecture Thu.2) Giustino, Lecture Tue.1 16/31
Temperature-dependent band structures |· · · | 2 � < 0 E c − E m m � = c Band gap |· · · | 2 � > 0 E v − E m m � = v Temperature ( Lecture Thu.2) Giustino, Lecture Tue.1 16/31
Temperature-dependent band structures silicon Figure from Zacharias et al, Phys. Rev. B 94, 075125 (2016) Giustino, Lecture Tue.1 17/31
Phonon-assisted optical absorption � m | ∂V SCF u | n � ∂τ � ∆ ψ n ( r ) = ψ m ( r ) E n − E m m � = n Giustino, Lecture Tue.1 18/31
Phonon-assisted optical absorption � m | ∂V SCF u | n � ∂τ � ∆ ψ n ( r ) = ψ m ( r ) E n − E m m � = n ǫ 2 ( ω ) = const p | v �| 2 δ ( E c − E v − � ω ) � |� c | ˆ ω 2 cv Giustino, Lecture Tue.1 18/31
Phonon-assisted optical absorption � m | ∂V SCF u | n � ∂τ � ∆ ψ n ( r ) = ψ m ( r ) E n − E m m � = n ǫ 2 ( ω ) = const p | v �| 2 δ ( E c − E v − � ω ) � |� c | ˆ ω 2 cv Giustino, Lecture Tue.1 18/31
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