ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First - - PowerPoint PPT Presentation

ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles

Trieste, 19-23 March 2018

Lecture Wed.1

Many-body theory of electron-phonon interactions

Feliciano Giustino

Department of Materials, University of Oxford Department of Materials Science and Engineering, Cornell University

Giustino, Lecture Wed.1 02/35

Lecture Summary

• Limitations of Rayleigh-Schr¨
• dinger perturbation theory
• Many-body Hamiltonian in quantum field theory
• Green’s function and the spectral function
• Electron-phonon self-energy
• Quasiparticle approximation
• Mass enhancement and electron lifetimes

Giustino, Lecture Wed.1 03/35

Limitations of Rayleigh-Schr¨

• dinger perturbation theory

Kohn-Sham equations again − 2 2me ∇2 ψn(r) + VSCF(r; τ1, τ2, · · ·) ψn(r) = En ψn(r)

Giustino, Lecture Wed.1 04/35

Limitations of Rayleigh-Schr¨

• dinger perturbation theory

Kohn-Sham equations again − 2 2me ∇2 ψn(r) + VSCF(r; τ1, τ2, · · ·) ψn(r) = En ψn(r)

• Adiabatic Born-Oppenheimer approximation

Giustino, Lecture Wed.1 04/35

Limitations of Rayleigh-Schr¨

• dinger perturbation theory

Kohn-Sham equations again − 2 2me ∇2 ψn(r) + VSCF(r; τ1, τ2, · · ·) ψn(r) = En ψn(r)

• Adiabatic Born-Oppenheimer approximation
• Nuclei described as classical particles

Giustino, Lecture Wed.1 04/35

Limitations of Rayleigh-Schr¨

• dinger perturbation theory

Kohn-Sham equations again − 2 2me ∇2 ψn(r) + VSCF(r; τ1, τ2, · · ·) ψn(r) = En ψn(r)

• Adiabatic Born-Oppenheimer approximation
• Nuclei described as classical particles
• Electron-phonon interactions depend on the XC functional

Giustino, Lecture Wed.1 04/35

Limitations of Rayleigh-Schr¨

• dinger perturbation theory

Kohn-Sham equations again − 2 2me ∇2 ψn(r) + VSCF(r; τ1, τ2, · · ·) ψn(r) = En ψn(r)

• Adiabatic Born-Oppenheimer approximation
• Nuclei described as classical particles
• Electron-phonon interactions depend on the XC functional
• Phonons are calculated from static displacements or DFPT

Giustino, Lecture Wed.1 04/35

Breakdown of Rayleigh-Schr¨

• dinger perturbation theory
• Polaron liquid at the SrTiO3(001) surface

Figure from Wang et al, Nature Mater. 15, 835 (2016)

Giustino, Lecture Wed.1 05/35

Breakdown of Rayleigh-Schr¨

• dinger perturbation theory
• Scanning tunneling spectra of 2H-NbS2

Figures from Guillam´

• n et al, Phys. Rev. Lett. 101, 166407 (2008)

and Heil et al, Phys. Rev. Lett. 119, 087003 (2017)

2∆

Giustino, Lecture Wed.1 06/35

Breakdown of Rayleigh-Schr¨

• dinger perturbation theory
• Raman G peak of gated graphene

Left figure from Pisana et al, Nat. Mater. 6, 198 (2007)

DFPT (RS) Giustino, Lecture Wed.1 07/35

Many-body Schr¨

• dinger’s equation

− 2 2me

• i∇2

i Ψ −

2 2Mκ

• κ∇2

κ Ψ −

• i,κ Zκv(ri, τκ) Ψ

+

• κ>κ′ ZκZκ′v(τκ, τκ′) Ψ +
• i>j v(ri, rj) Ψ = Etot Ψ

v(r, r′) = e2 4πǫ0|r − r′|

Giustino, Lecture Wed.1 08/35

Many-body Schr¨

• dinger’s equation

− 2 2me

• i∇2

i Ψ −

2 2Mκ

• κ∇2

κ Ψ −

• i,κ Zκv(ri, τκ) Ψ

+

• κ>κ′ ZκZκ′v(τκ, τκ′) Ψ +
• i>j v(ri, rj) Ψ = Etot Ψ

v(r, r′) = e2 4πǫ0|r − r′|

• We need to describe electrons and vibrations on the same footing

Giustino, Lecture Wed.1 08/35

Many-body Schr¨

• dinger’s equation

− 2 2me

• i∇2

i Ψ −

2 2Mκ

• κ∇2

κ Ψ −

• i,κ Zκv(ri, τκ) Ψ

+

• κ>κ′ ZκZκ′v(τκ, τκ′) Ψ +
• i>j v(ri, rj) Ψ = Etot Ψ

v(r, r′) = e2 4πǫ0|r − r′|

• We need to describe electrons and vibrations on the same footing
• The many-body Schr¨
• dinger equation is impractical for calculations

Giustino, Lecture Wed.1 08/35

Lattice Vibrations Electrons

Giustino, Lecture Wed.1 09/35

Lattice Vibrations Electrons

Giustino, Lecture Wed.1 09/35

Field operators

N-electron wavefunction as a linear combination of Slater determinants Ψ(x1, x2, · · · ) =

• mn

amn ˆ c†

mˆ

cn|0KS +

• mnpq

bmnpq ˆ c†

mˆ

c†

nˆ

cpˆ cq|0KS + · · ·

Giustino, Lecture Wed.1 10/35

Field operators

N-electron wavefunction as a linear combination of Slater determinants Ψ(x1, x2, · · · ) =

• mn

amn ˆ c†

mˆ

cn|0KS +

• mnpq

bmnpq ˆ c†

mˆ

c†

nˆ

cpˆ cq|0KS + · · · Operators in second quantization V (x1) + V (x2) + · · ·

• mn Vmnˆ

c†

mˆ

cn

Giustino, Lecture Wed.1 10/35

Field operators

N-electron wavefunction as a linear combination of Slater determinants Ψ(x1, x2, · · · ) =

• mn

amn ˆ c†

mˆ

cn|0KS +

• mnpq

bmnpq ˆ c†

mˆ

c†

nˆ

cpˆ cq|0KS + · · · Operators in second quantization V (x1) + V (x2) + · · ·

• mn Vmnˆ

c†

mˆ

cn =

• m
• n
• dxψ∗

m(x)V (x)ψn(x) ˆ

c†

mˆ

cn

Giustino, Lecture Wed.1 10/35

Field operators

N-electron wavefunction as a linear combination of Slater determinants Ψ(x1, x2, · · · ) =

• mn

amn ˆ c†

mˆ

cn|0KS +

• mnpq

bmnpq ˆ c†

mˆ

c†

nˆ

cpˆ cq|0KS + · · · Operators in second quantization V (x1) + V (x2) + · · ·

• mn Vmnˆ

c†

mˆ

cn =

• m
• n
• dxψ∗

m(x)V (x)ψn(x) ˆ

c†

mˆ

cn Field operators ˆ ψ(x) =

• n ψn(x) ˆ

cn

Giustino, Lecture Wed.1 10/35

Field operators

N-electron wavefunction as a linear combination of Slater determinants Ψ(x1, x2, · · · ) =

• mn

amn ˆ c†

mˆ

cn|0KS +

• mnpq

bmnpq ˆ c†

mˆ

c†

nˆ

cpˆ cq|0KS + · · · Operators in second quantization V (x1) + V (x2) + · · ·

• mn Vmnˆ

c†

mˆ

cn =

• m
• n
• dxψ∗

m(x)V (x)ψn(x) ˆ

c†

mˆ

cn =

• dx ˆ

ψ†(x)V (x) ˆ ψ(x) Field operators ˆ ψ(x) =

• n ψn(x) ˆ

cn

Giustino, Lecture Wed.1 10/35

Many-body Hamiltonian in second quantization

Non-relativistic Hamiltonian of coupled electrons and nuclei ˆ H = ˆ Te + ˆ Tn + ˆ Uen + ˆ Uee + ˆ Unn

Giustino, Lecture Wed.1 11/35

Many-body Hamiltonian in second quantization

Non-relativistic Hamiltonian of coupled electrons and nuclei ˆ H = ˆ Te + ˆ Tn + ˆ Uen + ˆ Uee + ˆ Unn Electron kinetic energy ˆ Te = − 2 2me

• dx ˆ

ψ†(x) ∇2 ˆ ψ(x)

Giustino, Lecture Wed.1 11/35

Many-body Hamiltonian in second quantization

Non-relativistic Hamiltonian of coupled electrons and nuclei ˆ H = ˆ Te + ˆ Tn + ˆ Uen + ˆ Uee + ˆ Unn Electron kinetic energy ˆ Te = − 2 2me

• dx ˆ

ψ†(x) ∇2 ˆ ψ(x) Electron-nucleus interaction ˆ Uen =

• dr
• dr′ ˆ

ne(r)ˆ nn(r′)v(r, r′), ˆ ne(r) =

• σ

ˆ ψ†(x) ˆ ψ(x)

Giustino, Lecture Wed.1 11/35

Many-body Hamiltonian in second quantization

Non-relativistic Hamiltonian of coupled electrons and nuclei ˆ H = ˆ Te + ˆ Tn + ˆ Uen + ˆ Uee + ˆ Unn Electron kinetic energy ˆ Te = − 2 2me

• dx ˆ

ψ†(x) ∇2 ˆ ψ(x) Electron-nucleus interaction ˆ Uen =

• dr
• dr′ ˆ

ne(r)ˆ nn(r′)v(r, r′), ˆ ne(r) =

• σ

ˆ ψ†(x) ˆ ψ(x) Electron-electron interaction ˆ Uee = 1 2

• dr
• dr′ ˆ

ne(r)

• ˆ

ne(r′) − δ(r − r′)

• v(r, r′)

Giustino, Lecture Wed.1 11/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Heisenberg time evolution

ˆ ψ(x, t) = ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/

i ∂ ∂t ˆ ψ(x, t) =

• ˆ

ψ(x, t), ˆ H

• Giustino, Lecture Wed.1

12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Heisenberg time evolution

ˆ ψ(x, t) = ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/

i ∂ ∂t ˆ ψ(x, t) =

• ˆ

ψ(x, t), ˆ H

• Exercise

N|ψ(x, t)|N + 1, s = N|ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/|N + 1, s

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Heisenberg time evolution

ˆ ψ(x, t) = ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/

i ∂ ∂t ˆ ψ(x, t) =

• ˆ

ψ(x, t), ˆ H

• Exercise

N|ψ(x, t)|N + 1, s = N|ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/|N + 1, s

= N|eiENt/ ˆ ψ(x) e−iEN+1,st/|N + 1, s

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Heisenberg time evolution

ˆ ψ(x, t) = ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/

i ∂ ∂t ˆ ψ(x, t) =

• ˆ

ψ(x, t), ˆ H

• Exercise

N|ψ(x, t)|N + 1, s = N|ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/|N + 1, s

= N|eiENt/ ˆ ψ(x) e−iEN+1,st/|N + 1, s = N| ˆ ψ(x)|N + 1, s e−iεst/

Giustino, Lecture Wed.1 12/35

Time evolution of field operators

Ground state of N-electron system ˆ H|N = EN|N s-th excited state of N +1-electron system ˆ H|N + 1, s = EN+1,s|N + 1, s Excitation energy εs = EN+1,s − EN

Heisenberg time evolution

ˆ ψ(x, t) = ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/

i ∂ ∂t ˆ ψ(x, t) =

• ˆ

ψ(x, t), ˆ H

• Exercise

N|ψ(x, t)|N + 1, s = N|ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/|N + 1, s

= N|eiENt/ ˆ ψ(x) e−iEN+1,st/|N + 1, s = N| ˆ ψ(x)|N + 1, s

• fs(x)

e−iεst/ Dyson orbital

Giustino, Lecture Wed.1 12/35

The Green’s function at zero temperature G(xt, x′t′) = − i N| ˆ T ˆ ψ(xt) ˆ ψ†(x′t′)|N

Time-ordered Green’s function Wick’s time-ordering operator

Giustino, Lecture Wed.1 13/35

The Green’s function at zero temperature G(xt, x′t′) = − i N| ˆ T ˆ ψ(xt) ˆ ψ†(x′t′)|N

Time-ordered Green’s function Wick’s time-ordering operator

• electron in x′ at time t′

Giustino, Lecture Wed.1 13/35

The Green’s function at zero temperature G(xt, x′t′) = − i N| ˆ T ˆ ψ(xt) ˆ ψ†(x′t′)|N

Time-ordered Green’s function Wick’s time-ordering operator

• electron in x at time t
• electron in x′ at time t′

Giustino, Lecture Wed.1 13/35

The Green’s function at zero temperature G(xt, x′t′) = − i N| ˆ T ˆ ψ(xt) ˆ ψ†(x′t′)|N

Time-ordered Green’s function Wick’s time-ordering operator

• electron in x at time t
• electron in x′ at time t′
• x′t′

xt •

Giustino, Lecture Wed.1 13/35

The Green’s function at zero temperature

Consider t > t′ (electron injection) G(xt, x′t′) = − i N| ˆ ψ(xt) ˆ ψ†(x′t′)|N

Giustino, Lecture Wed.1 14/35

The Green’s function at zero temperature

Consider t > t′ (electron injection) G(xt, x′t′) = − i N| ˆ ψ(xt) ˆ ψ†(x′t′)|N = − i N| ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/ ei ˆ Ht′/ ˆ

ψ†(x′) e−i ˆ

Ht′/|N

Giustino, Lecture Wed.1 14/35

The Green’s function at zero temperature

Consider t > t′ (electron injection) G(xt, x′t′) = − i N| ˆ ψ(xt) ˆ ψ†(x′t′)|N = − i N| ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/ ei ˆ Ht′/ ˆ

ψ†(x′) e−i ˆ

Ht′/|N

= − i N| ˆ ψ(x) e−i( ˆ

H−EN)(t−t′)/ ˆ

ψ†(x′)|N

Giustino, Lecture Wed.1 14/35

The Green’s function at zero temperature

Consider t > t′ (electron injection) G(xt, x′t′) = − i N| ˆ ψ(xt) ˆ ψ†(x′t′)|N = − i N| ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/ ei ˆ Ht′/ ˆ

ψ†(x′) e−i ˆ

Ht′/|N

= − i N| ˆ ψ(x) e−i( ˆ

H−EN)(t−t′)/ ˆ

ψ†(x′)|N

• s|N + 1, sN + 1, s|

Giustino, Lecture Wed.1 14/35

The Green’s function at zero temperature

Consider t > t′ (electron injection) G(xt, x′t′) = − i N| ˆ ψ(xt) ˆ ψ†(x′t′)|N = − i N| ei ˆ

Ht/ ˆ

ψ(x) e−i ˆ

Ht/ ei ˆ Ht′/ ˆ

ψ†(x′) e−i ˆ

Ht′/|N

= − i N| ˆ ψ(x) e−i( ˆ

H−EN)(t−t′)/ ˆ

ψ†(x′)|N = − i

• sfs(x)f∗

s (x′)e−iεs(t−t′)/

• s|N + 1, sN + 1, s|

Giustino, Lecture Wed.1 14/35

The spectral function

After carrying out the same operation for t < t′ and Fourier transform G(x, x′, ω) =

• s

fs(x)f∗

s (x′)

ω − εs ∓ i0+ ∓ occ/unocc

Giustino, Lecture Wed.1 15/35

The spectral function

After carrying out the same operation for t < t′ and Fourier transform G(x, x′, ω) =

• s

fs(x)f∗

s (x′)

ω − εs ∓ i0+ ∓ occ/unocc The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system

Giustino, Lecture Wed.1 15/35

The spectral function

After carrying out the same operation for t < t′ and Fourier transform G(x, x′, ω) =

• s

fs(x)f∗

s (x′)

ω − εs ∓ i0+ ∓ occ/unocc The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A(x, ω) = 1 π |Im G(x, x, ω)| =

• s |fs(x)|2 δ(ω − εs)

Giustino, Lecture Wed.1 15/35

The spectral function

After carrying out the same operation for t < t′ and Fourier transform G(x, x′, ω) =

• s

fs(x)f∗

s (x′)

ω − εs ∓ i0+ ∓ occ/unocc The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A(x, ω) = 1 π |Im G(x, x, ω)| =

• s |fs(x)|2 δ(ω − εs)

The spectra function is the many-body (local) density of states

Giustino, Lecture Wed.1 15/35

The spectral function

After carrying out the same operation for t < t′ and Fourier transform G(x, x′, ω) =

• s

fs(x)f∗

s (x′)

ω − εs ∓ i0+ ∓ occ/unocc The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A(x, ω) = 1 π |Im G(x, x, ω)| =

• s |fs(x)|2 δ(ω − εs)

The spectra function is the many-body (local) density of states

• Usually it is presented as momentum-resolved A(k, ω)

Giustino, Lecture Wed.1 15/35

The spectral function

Example: a single complex pole εs = ε − iΓ

Giustino, Lecture Wed.1 16/35

The spectral function

Example: a single complex pole εs = ε − iΓ G(x, x, t−t′) = − i |fs(x)|2 e−iε(t−t′)/e−Γ(t−t′)/

Giustino, Lecture Wed.1 16/35

The spectral function

Example: a single complex pole εs = ε − iΓ G(x, x, t−t′) = − i |fs(x)|2 e−iε(t−t′)/e−Γ(t−t′)/ |G(x, x, t−t′)| = 1 |fs(x)|2 e−Γ(t−t′)/

Giustino, Lecture Wed.1 16/35

The spectral function

Example: a single complex pole εs = ε − iΓ G(x, x, t−t′) = − i |fs(x)|2 e−iε(t−t′)/e−Γ(t−t′)/ |G(x, x, t−t′)| = 1 |fs(x)|2 e−Γ(t−t′)/

• decay

Giustino, Lecture Wed.1 16/35

The spectral function

Example: a single complex pole εs = ε − iΓ G(x, x, t−t′) = − i |fs(x)|2 e−iε(t−t′)/e−Γ(t−t′)/ |G(x, x, t−t′)| = 1 |fs(x)|2 e−Γ(t−t′)/

• decay

A(x, x, ω) = 1 π Γ (ω − ε)2 + Γ2 |fs(x)|2

Giustino, Lecture Wed.1 16/35

The spectral function

A(k, ω) k ω ε Γ

Example: a single complex pole εs = ε − iΓ G(x, x, t−t′) = − i |fs(x)|2 e−iε(t−t′)/e−Γ(t−t′)/ |G(x, x, t−t′)| = 1 |fs(x)|2 e−Γ(t−t′)/

• decay

A(x, x, ω) = 1 π Γ (ω − ε)2 + Γ2 |fs(x)|2

Giustino, Lecture Wed.1 16/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

Giustino, Lecture Wed.1 17/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

energy DFT density of states

Giustino, Lecture Wed.1 17/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

energy DFT density of states

many-body DOS

Giustino, Lecture Wed.1 17/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

energy DFT density of states

many-body DOS

quasiparticle shift

Giustino, Lecture Wed.1 17/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

energy DFT density of states

many-body DOS

quasiparticle shift quasiparticle broadening

Giustino, Lecture Wed.1 17/35

The spectral function

A(k, ω) = 1 π |Im G(k, ω)|

energy DFT density of states

many-body DOS

quasiparticle shift quasiparticle broadening boson energy

Giustino, Lecture Wed.1 17/35

How to calculate the Green’s function

Equation of motion for field operators i ∂ ∂t ˆ ψ(xt) =

• ˆ

ψ(x, t), ˆ H

• Giustino, Lecture Wed.1

18/35

How to calculate the Green’s function

Equation of motion for field operators i ∂ ∂t ˆ ψ(xt) =

• ˆ

ψ(x, t), ˆ H

• =
• − 2

2me ∇2 +

• dr′v(r, r′) ˆ

n(r′t)

• ˆ

ψ(xt) total charge, electrons & nuclei

Giustino, Lecture Wed.1 18/35

How to calculate the Green’s function

Equation of motion for field operators i ∂ ∂t ˆ ψ(xt) =

• ˆ

ψ(x, t), ˆ H

• =
• − 2

2me ∇2 +

• dr′v(r, r′) ˆ

n(r′t)

• ˆ

ψ(xt) total charge, electrons & nuclei i ∂ ∂t1 ˆ ψ(1) =

• − 2

2me ∇2

1 +

• d2v(12) ˆ

n(2)

• ˆ

ψ(1)

Giustino, Lecture Wed.1 18/35

How to calculate the Green’s function

Equation of motion for field operators i ∂ ∂t ˆ ψ(xt) =

• ˆ

ψ(x, t), ˆ H

• =
• − 2

2me ∇2 +

• dr′v(r, r′) ˆ

n(r′t)

• ˆ

ψ(xt) total charge, electrons & nuclei i ∂ ∂t1 ˆ ψ(1) =

• − 2

2me ∇2

1 +

• d2v(12) ˆ

n(2)

• ˆ

ψ(1) Equation of motion for Green’s function

• i ∂

∂t1 + 2 2me ∇2

1

• G(12) + i
• d3 v(13) ˆ

T ˆ n(3) ψ(1) ψ†(2) = δ(12)

Giustino, Lecture Wed.1 18/35

How to calculate the Green’s function

Equation of motion for field operators i ∂ ∂t ˆ ψ(xt) =

• ˆ

ψ(x, t), ˆ H

• =
• − 2

2me ∇2 +

• dr′v(r, r′) ˆ

n(r′t)

• ˆ

ψ(xt) total charge, electrons & nuclei i ∂ ∂t1 ˆ ψ(1) =

• − 2

2me ∇2

1 +

• d2v(12) ˆ

n(2)

• ˆ

ψ(1) Equation of motion for Green’s function

• i ∂

∂t1 + 2 2me ∇2

1

• G(12) + i
• d3 v(13) ˆ

T ˆ n(3) ψ(1) ψ†(2) = δ(12) 4 field operators → 2-particle Green’s function ˆ Tψ†(3)ψ(3)ψ(1)ψ†(2) = [Hartree] + [Fock] + G2(31, 32)

Giustino, Lecture Wed.1 18/35

How to calculate the Green’s function

• i ∂

∂t1 + 2 2me ∇2

1 − Vtot(1)

• G(12) −
• d3 Σ(13) G(32) = δ(12)

Vtot(1) =

• d2 v(12)ˆ

n(2) rewrite 2-particle Green’s function using self-energy Σ

Giustino, Lecture Wed.1 19/35

How to calculate the Green’s function

• i ∂

∂t1 + 2 2me ∇2

1 − Vtot(1)

• G(12) −
• d3 Σ(13) G(32) = δ(12)

Vtot(1) =

• d2 v(12)ˆ

n(2) rewrite 2-particle Green’s function using self-energy Σ Express the Green’s function in terms of Dyson’s orbitals

• − 2

2me ∇2 + Vtot(r)

• fs(x) +
• dx′ Σ(x, x′, εs/) fs(x′) = εsfs(x)

Giustino, Lecture Wed.1 19/35

How to calculate the Green’s function

• i ∂

∂t1 + 2 2me ∇2

1 − Vtot(1)

• G(12) −
• d3 Σ(13) G(32) = δ(12)

Vtot(1) =

• d2 v(12)ˆ

n(2) rewrite 2-particle Green’s function using self-energy Σ Sources of electron-phonon interaction Express the Green’s function in terms of Dyson’s orbitals

• − 2

2me ∇2 + Vtot(r)

• fs(x) +
• dx′ Σ(x, x′, εs/) fs(x′) = εsfs(x)

Giustino, Lecture Wed.1 19/35

How to calculate the Green’s function

• i ∂

∂t1 + 2 2me ∇2

1 − Vtot(1)

• G(12) −
• d3 Σ(13) G(32) = δ(12)

Vtot(1) =

• d2 v(12)ˆ

n(2) rewrite 2-particle Green’s function using self-energy Σ Sources of electron-phonon interaction Express the Green’s function in terms of Dyson’s orbitals

• − 2

2me ∇2 + Vtot(r)

• fs(x) +
• dx′ Σ(x, x′, εs/) fs(x′) = εsfs(x)

Giustino, Lecture Wed.1 19/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Green’s function

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Green’s function Vertex

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Green’s function Vertex Screened Coulomb interaction

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Green’s function Vertex Screened Coulomb interaction

W = We + Wph

We(12) =

• d3 ǫ−1

e (13)v(32)

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Electron self-energy from Hedin-Baym’s equations

Σ(12) = i

• d(34) G(13) Γ(324)W(41+)

Green’s function Vertex Screened Coulomb interaction

W = We + Wph

We(12) =

• d3 ǫ−1

e (13)v(32)

Basically the standard GW method + screening from nuclei

Giustino, Lecture Wed.1 20/35

How to calculate the self-energy

Screened Coulomb interaction from the nuclei Wph(12) =

• κκ′
• d(34) ǫ−1

e (13)∂Vκ(r3)

∂τκ · Dκκ′(t3t4) · ǫ−1

e (24)∂Vκ′(r4)

∂τκ′

Giustino, Lecture Wed.1 21/35

How to calculate the self-energy

“electron-phonon matrix elements”

Screened Coulomb interaction from the nuclei Wph(12) =

• κκ′
• d(34) ǫ−1

e (13)∂Vκ(r3)

∂τκ · Dκκ′(t3t4) · ǫ−1

e (24)∂Vκ′(r4)

∂τκ′

Giustino, Lecture Wed.1 21/35

How to calculate the self-energy

“electron-phonon matrix elements”

Screened Coulomb interaction from the nuclei Wph(12) =

• κκ′
• d(34) ǫ−1

e (13)∂Vκ(r3)

∂τκ · Dκκ′(t3t4) · ǫ−1

e (24)∂Vκ′(r4)

∂τκ′ Displacement-displacement correlation function of the nuclei, a.k.a. the phonon Green’s function Dκκ′(tt′) = − i ˆ T ∆ˆ τκ(t) ∆ˆ τ T

κ′(t′)

Giustino, Lecture Wed.1 21/35

Diagrammatic representation of the self-energy

Figure from Giustino,

• Rev. Mod. Phys. 89,

015003 (2017)

Standard GW self-energy

(we will ignore this from now on)

Giustino, Lecture Wed.1 22/35

Diagrammatic representation of the self-energy

Figure from Giustino,

• Rev. Mod. Phys. 89,

015003 (2017)

Standard GW self-energy

(we will ignore this from now on)

Fan-Migdal self-energy

Giustino, Lecture Wed.1 22/35

Diagrammatic representation of the self-energy

Figure from Giustino,

• Rev. Mod. Phys. 89,

015003 (2017)

Standard GW self-energy

(we will ignore this from now on)

Fan-Migdal self-energy Debye-Waller self-energy (Lecture Thu.2)

Improper self-energy: comes form Vtot(1) =

• d2 v(12)ˆ

n(2) term

Giustino, Lecture Wed.1 22/35

Diagrammatic representation of the self-energy

Figure from Giustino,

• Rev. Mod. Phys. 89,

015003 (2017)

Standard GW self-energy

(we will ignore this from now on)

Fan-Migdal self-energy Debye-Waller self-energy (Lecture Thu.2)

Improper self-energy: comes form Vtot(1) =

• d2 v(12)ˆ

n(2) term

Giustino, Lecture Wed.1 22/35

Fan-Migdal self-energy

Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons ΣFM

nk (ω) = 1

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• 1 − fmk+q

ω−εmk+q/ − ωqν + iη + fmk+q ω−εmk+q/ + ωqν + iη

• Giustino, Lecture Wed.1

23/35

Fan-Migdal self-energy

Dynamical structure on the scale

• f the phonon energy

Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons ΣFM

nk (ω) = 1

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• 1 − fmk+q

ω−εmk+q/ − ωqν + iη + fmk+q ω−εmk+q/ + ωqν + iη

• Giustino, Lecture Wed.1

23/35

Fan-Migdal self-energy

Dynamical structure on the scale

• f the phonon energy

Summation over all phonon branches and wavevectors

Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons ΣFM

nk (ω) = 1

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• 1 − fmk+q

ω−εmk+q/ − ωqν + iη + fmk+q ω−εmk+q/ + ωqν + iη

• Giustino, Lecture Wed.1

23/35

Fan-Migdal self-energy

Dynamical structure on the scale

• f the phonon energy

Summation over all phonon branches and wavevectors Extension to finite temperature

Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons ΣFM

nk (ω) = 1

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• 1 − fmk+q + nqν

ω−εmk+q/ − ωqν + iη + fmk+q + nqν ω−εmk+q/ + ωqν + iη

• Giustino, Lecture Wed.1

23/35

Fan-Migdal self-energy

εF Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Fan-Migdal self-energy

εF

Wavevector Energy Wavevector E n e r g y

Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Fan-Migdal self-energy

εF

Wavevector Energy Wavevector E n e r g y

Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Fan-Migdal self-energy

εF

Wavevector Energy Wavevector E n e r g y

Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Fan-Migdal self-energy

εF

Wavevector Energy Wavevector E n e r g y

phonon energy

Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Fan-Migdal self-energy

εF

Wavevector Energy Wavevector E n e r g y

phonon energy broadening change of velocity/mass

Example: A single dispersionless phonon (Holstein model)

Giustino, Lecture Wed.1 24/35

Examples from experiments

• Velocity renormalization in MgB2

Right figure from Mou et al, Phys. Rev. B 91, 140502(R) (2015)

v = v0/2.4

Giustino, Lecture Wed.1 25/35

Examples from experiments

• Velocity renormalization in Ca-decorated graphene on Au

Right figure adapted from Fedorov et al, Nat. Commun. 5, 3257 (2014)

v = v0/1.25

Giustino, Lecture Wed.1 26/35

Examples from calculations

• Velocity renormalization in C6CaC6 (EPW)

Figure adapted from Margine et al, Sci Rep. 6, 21414 (2016)

Giustino, Lecture Wed.1 27/35

Examples from calculations

• Velocity renormalization and broadening in MgB2

Figure from Eiguren et al, Phys. Rev. B 79. 245103 (2009)

Giustino, Lecture Wed.1 28/35

Quasiparticle shift and broadening

Spectral function from the self-energy A(k, ω) = − 1 π Im

• n

1 ω−εnk−Σnk(ω)

Giustino, Lecture Wed.1 29/35

Quasiparticle shift and broadening

Spectral function from the self-energy A(k, ω) = − 1 π Im

• n

1 ω−εnk−Σnk(ω) Quasiparticle approximation: assume Lorentzian peaks centered near ω = Enk Σnk(ω) = Σnk(Enk) + 1

• ∂ReΣnk

∂ω

• ω=Enk/

(ω − Enk) + · · ·

Giustino, Lecture Wed.1 29/35

Quasiparticle shift and broadening

Spectral function from the self-energy A(k, ω) = − 1 π Im

• n

1 ω−εnk−Σnk(ω) Quasiparticle approximation: assume Lorentzian peaks centered near ω = Enk Σnk(ω) = Σnk(Enk) + 1

• ∂ReΣnk

∂ω

• ω=Enk/

(ω − Enk) + · · · Define the quasiparticle strength Znk =

• 1 − 1
• ∂ReΣnk(ω)

∂ω

• ω=Enk/
• −1

Giustino, Lecture Wed.1 29/35

Quasiparticle shift and broadening

Replace the Taylor expansion inside the spectral function A(k, ω) = − 1 π

• n

1 ω−εnk−Σnk(Enk) − (1 − 1/Znk)(ω − Enk)

Giustino, Lecture Wed.1 30/35

Quasiparticle shift and broadening

Replace the Taylor expansion inside the spectral function A(k, ω) = − 1 π

• n

1 ω−εnk−Σnk(Enk) − (1 − 1/Znk)(ω − Enk) After rearranging(∗): A(k, ω) = − 1 π

• n

Znk ω− (Enk + iΓnk)

(∗)Requires the additional approximation |∂ImΣnk/∂ω| ≪ |∂ReΣnk/∂ω| Giustino, Lecture Wed.1 30/35

Quasiparticle shift and broadening

Replace the Taylor expansion inside the spectral function A(k, ω) = − 1 π

• n

1 ω−εnk−Σnk(Enk) − (1 − 1/Znk)(ω − Enk) After rearranging(∗): A(k, ω) = − 1 π

• n

Znk ω− (Enk + iΓnk) Enk = εnk + Re Σnk(Enk/)

quasiparticle energy

(∗)Requires the additional approximation |∂ImΣnk/∂ω| ≪ |∂ReΣnk/∂ω| Giustino, Lecture Wed.1 30/35

Quasiparticle shift and broadening

Replace the Taylor expansion inside the spectral function A(k, ω) = − 1 π

• n

1 ω−εnk−Σnk(Enk) − (1 − 1/Znk)(ω − Enk) After rearranging(∗): A(k, ω) = − 1 π

• n

Znk ω− (Enk + iΓnk) Enk = εnk + Re Σnk(Enk/) Γnk = Znk Im Σnk(Enk/)

quasiparticle energy quasiparticle broadening

(∗)Requires the additional approximation |∂ImΣnk/∂ω| ≪ |∂ReΣnk/∂ω| Giustino, Lecture Wed.1 30/35

The mass enhancement parameter

Taking the k-derivatives of the quasiparticle energy Enk we find the velocity and mass renormalization Vnk = vnk 1 + λnk M∗

nk = (1 + λnk) m∗ nk

(valid only for simple metals)

Giustino, Lecture Wed.1 31/35

The mass enhancement parameter

Taking the k-derivatives of the quasiparticle energy Enk we find the velocity and mass renormalization Vnk = vnk 1 + λnk M∗

nk = (1 + λnk) m∗ nk

(valid only for simple metals)

λnk is the mass enhancement parameter λnk = 1 Znk − 1

Giustino, Lecture Wed.1 31/35

The mass enhancement parameter

Taking the k-derivatives of the quasiparticle energy Enk we find the velocity and mass renormalization Vnk = vnk 1 + λnk M∗

nk = (1 + λnk) m∗ nk

(valid only for simple metals)

λnk is the mass enhancement parameter λnk = 1 Znk − 1 = −1

• ∂ ReΣnk(ω)

∂ω

• ω=Enk/

Giustino, Lecture Wed.1 31/35

The mass enhancement parameter

Taking the k-derivatives of the quasiparticle energy Enk we find the velocity and mass renormalization Vnk = vnk 1 + λnk M∗

nk = (1 + λnk) m∗ nk

(valid only for simple metals)

λnk is the mass enhancement parameter λnk = 1 Znk − 1 = −1

• ∂ ReΣnk(ω)

∂ω

• ω=Enk/

= −1

• ∂ ImΣnk(ω)

∂η

• ω=Enk/

(Cauchy-Riemann condition)

Giustino, Lecture Wed.1 31/35

Electron lifetimes

τnk =

• 2Γnk

=

• 2 |ZnkIm Σnk(Enk/)|

Giustino, Lecture Wed.1 32/35

Electron lifetimes

τnk =

• 2Γnk

=

• 2 |ZnkIm Σnk(Enk/)|

Common approximation: replace Enk by εnk and set Znk = 1

Giustino, Lecture Wed.1 32/35

Electron lifetimes

τnk =

• 2Γnk

=

• 2 |ZnkIm Σnk(Enk/)|

Common approximation: replace Enk by εnk and set Znk = 1 1 τnk = 2π

• mν

dq ΩBZ |gnmν(k, q)|2 × [(1 − fmk+q + nqν)δ(εnk − ωqν −εmk+q) + (fmk+q + nqν)δ(εnk + ωqν −εmk+q)]

Giustino, Lecture Wed.1 32/35

Electron lifetimes

τnk =

• 2Γnk

=

• 2 |ZnkIm Σnk(Enk/)|

Common approximation: replace Enk by εnk and set Znk = 1 1 τnk = 2π

• mν

dq ΩBZ |gnmν(k, q)|2 × [(1 − fmk+q + nqν)δ(εnk − ωqν −εmk+q) + (fmk+q + nqν)δ(εnk + ωqν −εmk+q)] phonon emission phonon absorption

Giustino, Lecture Wed.1 32/35

Electron lifetimes

τnk =

• 2Γnk

=

• 2 |ZnkIm Σnk(Enk/)|

Common approximation: replace Enk by εnk and set Znk = 1 1 τnk = 2π

• mν

dq ΩBZ |gnmν(k, q)|2 × [(1 − fmk+q + nqν)δ(εnk − ωqν −εmk+q) + (fmk+q + nqν)δ(εnk + ωqν −εmk+q)] phonon emission phonon absorption Standard Fermi Golden rule expression for lifetimes

Giustino, Lecture Wed.1 32/35

Example from calculations

• Electron lifetimes in anatase TiO2 (EPW)

Figure adapted from Verdi et al, Phys. Rev. Lett. 115, 176401 (2015)

Giustino, Lecture Wed.1 33/35

Take-home messages

• Quantum field theory is extremely useful in the study of

electron-phonon physics

• The electron-phonon self-energy works as in the GW

method, but on much smaller energy scales

• We can calculate the change of the effective mass and

band velocity induced by EPIs

• We can calculate electron lifetimes arising from EPIs

Giustino, Lecture Wed.1 34/35

References

• F. Giustino, Rev. Mod. Phys. 89, 015003 (2017)

[link]

• A. Marini, S. Ponc´

e, and X. Gonze, Phys. Rev. B 91, 224310 (2015) [Link]

• A. Eiguren, C. Ambrosch-Draxl, and P. M. Echenique, Phys. Rev. B 79,

245103 (2009) [Link]

• Abrikosov et al, Methods of quantum field theory in statistical physics,

1964

• L. Hedin and S. Lundqvist, Effects of electron-electron and

electron-phonon interactions on the one-electron states of solids, Ed. Seitz, Turnbull, and Ehrenreich, Solid State Physics, Vol. 23 (Academic, 1969)

• G. Grimvall, The electron-phonon interaction in metals, 1981,

(North-Holland, Amsterdam)

• S. Engelsberg and J. R. Schrieffer, Phys. Rev. 131, 993 (1963)

[Link]

• G. D. Mahan, Many-Particle Physics (Plenum, 1993)

Giustino, Lecture Wed.1 35/35