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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

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

Trieste, 19-23 March 2018

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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

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Lecture Summary

  • Manifestations of the electron-phonon interaction
  • Rayleigh-Schr¨
  • dinger 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

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Where do electron-phonon interactions come from?

Giustino, Lecture Tue.1 04/31

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Ionic degrees of freedom in the Kohn-Sham equations

− 2 2me ∇2 ψn + VSCF ψn = En ψn

Giustino, Lecture Tue.1 05/31

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Ionic degrees of freedom in the Kohn-Sham equations

− 2 2me ∇2 ψn + VSCF ψn = En ψn n(r) =

  • n∈occ

|ψn(r)|2

Giustino, Lecture Tue.1 05/31

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Ionic degrees of freedom in the Kohn-Sham equations

− 2 2me ∇2 ψn + VSCF ψn = En ψn n(r) =

  • n∈occ

|ψn(r)|2 VSCF(r) = − e2 4πǫ0

κ

Zκ |r − τ κ| − n(r′)dr′ |r − r′|

  • + Vxc[n(r)]

Giustino, Lecture Tue.1 05/31

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SLIDE 8

Ionic degrees of freedom in the Kohn-Sham equations

− 2 2me ∇2 ψn + VSCF ψn = En ψn n(r) =

  • n∈occ

|ψn(r)|2 VSCF(r) = − e2 4πǫ0

κ

Zκ |r − τ κ| − n(r′)dr′ |r − r′|

  • + Vxc[n(r)]

Giustino, Lecture Tue.1 05/31

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Ionic degrees of freedom in the Kohn-Sham equations

The SCF potential depends parametrically on the ionic coordinates

VSCF(r; τ 1, τ 2, · · · )

Giustino, Lecture Tue.1 06/31

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Ionic degrees of freedom in the Kohn-Sham equations

The SCF potential depends parametrically on the ionic coordinates

VSCF(r; τ 1, τ 2, · · · )

  • Consider only one ion and one Cartesian direction for simplicity
  • Displace atoms from equilibrium sites, τ = τ0 + u

Giustino, Lecture Tue.1 06/31

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Ionic degrees of freedom in the Kohn-Sham equations

The SCF potential depends parametrically on the ionic coordinates

VSCF(r; τ 1, τ 2, · · · )

  • Consider only one ion and one Cartesian direction for simplicity
  • Displace atoms from equilibrium sites, τ = τ0 + u

VSCF(r; τ) = VSCF(r; τ0) + ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

Giustino, Lecture Tue.1 06/31

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Ionic degrees of freedom in the Kohn-Sham equations

The SCF potential depends parametrically on the ionic coordinates

VSCF(r; τ 1, τ 2, · · · )

  • Consider only one ion and one Cartesian direction for simplicity
  • Displace atoms from equilibrium sites, τ = τ0 + u

Perturbation Hamiltonian leading to EPIs

VSCF(r; τ) = VSCF(r; τ0) + ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

Giustino, Lecture Tue.1 06/31

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Some manifestations of electron-phonon interactions

  • Electron mobility in monolayer and bilayer MoS2

Figure from Baugher et al, Nano Lett. 13, 4212 (2013)

Giustino, Lecture Tue.1 07/31

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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

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Some manifestations of electron-phonon interactions

  • High-temperature superconductivity in compressed H3S

Figure from Drozdov et al, Nature 73, 525 (2015)

Giustino, Lecture Tue.1 09/31

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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

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Some manifestations of electron-phonon interactions

  • Electron mass enhancement in MgB2

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

Giustino, Lecture Tue.1 11/31

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Rayleigh-Schr¨

  • dinger perturbation theory

∆ ˆ Hep = ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

Giustino, Lecture Tue.1 12/31

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Rayleigh-Schr¨

  • dinger perturbation theory

∆ ˆ Hep = ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

  • Energies

∆En = n| ∂VSCF ∂τ u |n

Giustino, Lecture Tue.1 12/31

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Rayleigh-Schr¨

  • dinger perturbation theory

∆ ˆ Hep = ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

  • Energies

∆En = n| ∂VSCF ∂τ u |n

  • Wavefunctions

∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

Giustino, Lecture Tue.1 12/31

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Rayleigh-Schr¨

  • dinger perturbation theory

∆ ˆ Hep = ∂VSCF ∂τ u + 1 2 ∂2VSCF ∂τ 2 u2 + · · ·

  • Energies

∆En = n| ∂VSCF ∂τ u |n

  • Wavefunctions

∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

  • Transition rates

Γn→m = 2π |m| ∂VSCF ∂τ u |n|2 δ(Em−En−ω)

Giustino, Lecture Tue.1 12/31

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Thermodynamic averages

What is the atomic displacement u in ∆ ˆ Hep?

Giustino, Lecture Tue.1 13/31

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Thermodynamic averages

What is the atomic displacement u in ∆ ˆ Hep?

u M C = Mω2

Giustino, Lecture Tue.1 13/31

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Thermodynamic averages

What is the atomic displacement u in ∆ ˆ Hep?

u M C = Mω2 u2T = kBT Mω2

classical

Giustino, Lecture Tue.1 13/31

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Thermodynamic averages

What is the atomic displacement u in ∆ ˆ Hep?

u M C = Mω2 u2T = kBT Mω2

classical

u2T =

  • 2Mω
  • 2 n

ω kBT

  • + 1
  • classical

quantum

Giustino, Lecture Tue.1 13/31

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Thermodynamic averages

∆EnT −

− − − − − − − − → Temperature-dependent band structures

· · · ∆ψn(r) · · ·T −

− → Phonon-assisted optical absorption

Γn→mT −

− − − − − − − − → Phonon-limited carrier mobilities

Giustino, Lecture Tue.1 14/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u +

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em u2

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u +

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em u2 +1 2n|∂2VSCF ∂τ 2 |n u2

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u +

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em u2 +1 2n|∂2VSCF ∂τ 2 |n u2 ∆EnT =     

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em + 1 2n|∂2VSCF ∂τ 2 |n      u2T

(Lecture Thu.2)

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

∆En = n|∂VSCF ∂τ |n u +

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em u2 +1 2n|∂2VSCF ∂τ 2 |n u2 ∆EnT =     

  • m=n
  • m|∂VSCF

∂τ |n

  • 2

En − Em + 1 2n|∂2VSCF ∂τ 2 |n     

  • 2Mω (2 nT + 1)

(Lecture Thu.2)

Giustino, Lecture Tue.1 15/31

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Temperature-dependent band structures

  • m=c

|· · · |2 Ec − Em < 0

  • m=v

|· · · |2 Ev − Em > 0

(Lecture Thu.2)

Giustino, Lecture Tue.1 16/31

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Temperature-dependent band structures

Temperature Band gap

  • m=c

|· · · |2 Ec − Em < 0

  • m=v

|· · · |2 Ev − Em > 0

(Lecture Thu.2)

Giustino, Lecture Tue.1 16/31

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Temperature-dependent band structures silicon

Figure from Zacharias et al, Phys. Rev. B 94, 075125 (2016)

Giustino, Lecture Tue.1 17/31

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Phonon-assisted optical absorption

∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

Giustino, Lecture Tue.1 18/31

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Phonon-assisted optical absorption

ǫ2(ω) = const ω2

  • cv

|c| ˆ p |v|2 δ(Ec − Ev − ω) ∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

Giustino, Lecture Tue.1 18/31

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Phonon-assisted optical absorption

ǫ2(ω) = const ω2

  • cv

|c| ˆ p |v|2 δ(Ec − Ev − ω) ∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

Giustino, Lecture Tue.1 18/31

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Phonon-assisted optical absorption

ǫ2(ω) = const ω2

  • cv

|c| ˆ p |v|2 δ(Ec − Ev − ω) ∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

  • m=c

c| ∂VSCF ∂τ |mm| ˆ p |v Ec − Em + · · ·

  • 2

u2T

Giustino, Lecture Tue.1 18/31

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Phonon-assisted optical absorption

ǫ2(ω) = const ω2

  • cv

|c| ˆ p |v|2 δ(Ec − Ev − ω) ∆ψn(r) =

  • m=n

m| ∂VSCF ∂τ u |n En − Em ψm(r)

  • m=c

c| ∂VSCF ∂τ |mm| ˆ p |v Ec − Em + · · ·

  • 2

u2T wavevector energy (Lecture Fri.1)

Giustino, Lecture Tue.1 18/31

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Phonon-assisted optical absorption silicon

Figure from Zacharias et al, Phys. Rev. Lett. 115, 177401 (2015)

Giustino, Lecture Tue.1 19/31

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Phonon-assisted optical absorption silicon

Figure from Zacharias et al, Phys. Rev. Lett. 115, 177401 (2015)

Giustino, Lecture Tue.1 19/31

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Phonon-assisted optical absorption silicon

Figure from Zacharias et al, Phys. Rev. Lett. 115, 177401 (2015)

Giustino, Lecture Tue.1 19/31

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Phonon-limited carrier mobilities

Carrier relaxation time 1 τn =

  • m Γn → m

Giustino, Lecture Tue.1 20/31

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Phonon-limited carrier mobilities

Carrier relaxation time 1 τn =

  • m Γn → m

Electron mobility from Boltzmann equation (Lecture Wed.2) µ = e m 1 3 e−(En−EF)/kBT m|vn|2 kBT τn

  • CB

Giustino, Lecture Tue.1 20/31

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SLIDE 46

Phonon-limited carrier mobilities

Carrier relaxation time 1 τn =

  • m Γn → m

Electron mobility from Boltzmann equation (Lecture Wed.2) µ = e m 1 3 e−(En−EF)/kBT m|vn|2 kBT τn

  • CB

µ = eτ m Drude formula

Giustino, Lecture Tue.1 20/31

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The electron-phonon matrix element

Matrix element Zero-point displacement m|∂VSCF ∂τ |n u2T =

  • 2Mω

Giustino, Lecture Tue.1 21/31

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The electron-phonon matrix element

Matrix element Zero-point displacement m|∂VSCF ∂τ |n u2T =

  • 2Mω

electrons in GaAs phonons in GaAs

Giustino, Lecture Tue.1 21/31

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The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc

Giustino, Lecture Tue.1 22/31

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The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc

Lattice-periodic part of wavefunction

Giustino, Lecture Tue.1 22/31

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SLIDE 51

The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential

Giustino, Lecture Tue.1 22/31

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The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

  • καpe−iq·(r−Rp)
  • 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

Giustino, Lecture Tue.1 22/31

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SLIDE 53

The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

  • καpe−iq·(r−Rp)
  • 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential Zero-point amplitude κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

Giustino, Lecture Tue.1 22/31

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SLIDE 54

The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

  • καpe−iq·(r−Rp)
  • 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential Zero-point amplitude Phonon polarization κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

Giustino, Lecture Tue.1 22/31

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The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

  • καpe−iq·(r−Rp)
  • 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential Zero-point amplitude Phonon polarization Displacement of a single ion κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

Giustino, Lecture Tue.1 22/31

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The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

  • καpe−iq·(r−Rp)
  • 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential Zero-point amplitude Phonon polarization Displacement of a single ion Incommensurate modulation κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

Giustino, Lecture Tue.1 22/31

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SLIDE 57

Brillouin-zone integrals

Example: electron lifetimes in metals, adiabatic approximation 1 τnk = 2kBT 2π

  • mν
  • BZ

dq ΩBZ |gnmν(k, q)|2 ωqν δ(εnk − εmk+q)

Giustino, Lecture Tue.1 23/31

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SLIDE 58

Brillouin-zone integrals

Example: electron lifetimes in metals, adiabatic approximation 1 τnk = 2kBT 2π

  • mν
  • BZ

dq ΩBZ |gnmν(k, q)|2 ωqν δ(εnk − εmk+q)

  • The integral over the Brillouin zone can require up to 100K q-vectors

Giustino, Lecture Tue.1 23/31

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SLIDE 59

Brillouin-zone integrals

Example: electron lifetimes in metals, adiabatic approximation 1 τnk = 2kBT 2π

  • mν
  • BZ

dq ΩBZ |gnmν(k, q)|2 ωqν δ(εnk − εmk+q)

  • The integral over the Brillouin zone can require up to 100K q-vectors
  • Each q-vector requires a separate DFPT calculation

Giustino, Lecture Tue.1 23/31

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SLIDE 60

Brillouin-zone integrals

Example: electron lifetimes in metals, adiabatic approximation 1 τnk = 2kBT 2π

  • mν
  • BZ

dq ΩBZ |gnmν(k, q)|2 ωqν δ(εnk − εmk+q)

  • The integral over the Brillouin zone can require up to 100K q-vectors
  • Each q-vector requires a separate DFPT calculation
  • A new integral must be evaluated for every k-vector

Giustino, Lecture Tue.1 23/31

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SLIDE 61

Wannier interpolation of electron-phonon matrix elements

Wannier functions (Lecture Tue.2) wmp(r) = 1 Np

  • nk

e−ik·Rp Unmk ψnk(r),

Giustino, Lecture Tue.1 24/31

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SLIDE 62

Wannier interpolation of electron-phonon matrix elements

Wannier functions (Lecture Tue.2) wmp(r) = 1 Np

  • nk

e−ik·Rp Unmk ψnk(r), ψnk(r)

Giustino, Lecture Tue.1 24/31

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SLIDE 63

Wannier interpolation of electron-phonon matrix elements

Wannier functions (Lecture Tue.2) wmp(r) = 1 Np

  • nk

e−ik·Rp Unmk ψnk(r), ψnk(r) wmp(r)

Giustino, Lecture Tue.1 24/31

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SLIDE 64

Wannier interpolation of electron-phonon matrix elements

w2(r) w1(r) ∂VSCF ∂τκαp (r)

Giustino, Lecture Tue.1 25/31

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SLIDE 65

Wannier interpolation of electron-phonon matrix elements

w2(r) w1(r) ∂VSCF ∂τκαp (r) gmnν(k, q) =

  • 2Mκωqν
  • pp′

ei(k·Rp+q·Rp′) Uk+q g(Rp, Rp′) · eqν U †

k

  • mn

(Lecture Wed.3)

Giustino, Lecture Tue.1 25/31

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SLIDE 66

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS

Giustino, Lecture Tue.1 26/31

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SLIDE 67

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

Giustino, Lecture Tue.1 26/31

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SLIDE 68

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

· · · FS is the average over all possible combinations with εnk and εmk+q on the Fermi surface

Giustino, Lecture Tue.1 26/31

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SLIDE 69

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

· · · FS is the average over all possible combinations with εnk and εmk+q on the Fermi surface

  • Dimensionless, typically ranges between 0 and 2

Giustino, Lecture Tue.1 26/31

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SLIDE 70

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

· · · FS is the average over all possible combinations with εnk and εmk+q on the Fermi surface

  • Dimensionless, typically ranges between 0 and 2
  • In BCS theory the critical temperature is ∝ exp(−1/λ)

Giustino, Lecture Tue.1 26/31

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SLIDE 71

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

· · · FS is the average over all possible combinations with εnk and εmk+q on the Fermi surface

  • Dimensionless, typically ranges between 0 and 2
  • In BCS theory the critical temperature is ∝ exp(−1/λ)
  • In metals the electron mass enhancement is 1 + λ

Giustino, Lecture Tue.1 26/31

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SLIDE 72

The electron-phonon coupling constant

λ = NF

  • ν

|gmnν(k, q)|2 ωqν

  • FS
  • Defined for metals

· · · FS is the average over all possible combinations with εnk and εmk+q on the Fermi surface

  • Dimensionless, typically ranges between 0 and 2
  • In BCS theory the critical temperature is ∝ exp(−1/λ)
  • In metals the electron mass enhancement is 1 + λ
  • Not meaningful for intrinsic semiconductors and insulators

(Lecture Wed.1)

Giustino, Lecture Tue.1 26/31

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SLIDE 73

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger
  • Time-evolution of DFT band gap of CH3NH3PbI3

Right figure from Quarti et al, Phys. Chem. Chem. Phys. 17, 9394 (2015)

Giustino, Lecture Tue.1 27/31

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SLIDE 74

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

We have seen that DFT eigenvalues depend parametrically on the ionic displacements En(u) = En(0) + C1 u + C2 u2 + · · ·

Giustino, Lecture Tue.1 28/31

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SLIDE 75

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

We have seen that DFT eigenvalues depend parametrically on the ionic displacements En(u) = En(0) + C1 u + C2 u2 + · · ·

  • In RS perturbation theory we determine C2 perturbatively,

and we take the (classical or quantum) thermal averages of the displacements, u2T

Giustino, Lecture Tue.1 28/31

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SLIDE 76

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

We have seen that DFT eigenvalues depend parametrically on the ionic displacements En(u) = En(0) + C1 u + C2 u2 + · · ·

  • In RS perturbation theory we determine C2 perturbatively,

and we take the (classical or quantum) thermal averages of the displacements, u2T

  • In MD simulations (classical or Path Integral) we evaluate

explicitly En(u) and we average over MD snapshots, En(u)T

Giustino, Lecture Tue.1 28/31

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SLIDE 77

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

We have seen that DFT eigenvalues depend parametrically on the ionic displacements En(u) = En(0) + C1 u + C2 u2 + · · ·

  • In RS perturbation theory we determine C2 perturbatively,

and we take the (classical or quantum) thermal averages of the displacements, u2T

  • In MD simulations (classical or Path Integral) we evaluate

explicitly En(u) and we average over MD snapshots, En(u)T

  • These two approaches are equivalent for harmonic systems

Giustino, Lecture Tue.1 28/31

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SLIDE 78

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

Probability distribution of ionic displacements (harmonic system)

displacement u energy χ1 χ2 χ3

Giustino, Lecture Tue.1 29/31

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SLIDE 79

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

Probability distribution of ionic displacements (harmonic system) Prob (u) = 1 Z

∞

  • n=0

e−(n+1/2)ω/kBT |χn(u)|2

displacement u energy χ1 χ2 χ3

Giustino, Lecture Tue.1 29/31

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SLIDE 80

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

Probability distribution of ionic displacements (harmonic system) Prob (u) = 1 Z

∞

  • n=0

e−(n+1/2)ω/kBT |χn(u)|2 = 1

  • 2πu2T

exp

  • −

u2 2u2T

  • Meheler formula

displacement u energy χ1 χ2 χ3

Giustino, Lecture Tue.1 29/31

slide-81
SLIDE 81

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

Probability distribution of ionic displacements (harmonic system) Prob (u) = 1 Z

∞

  • n=0

e−(n+1/2)ω/kBT |χn(u)|2 = 1

  • 2πu2T

exp

  • −

u2 2u2T

  • Meheler formula

displacement u energy χ1 χ2 χ3 displacement u Prob (u) T

Giustino, Lecture Tue.1 29/31

slide-82
SLIDE 82

Molecular Dynamics vs. Rayleigh-Schr¨

  • dinger

Probability distribution of ionic displacements (harmonic system) Prob (u) = 1 Z

∞

  • n=0

e−(n+1/2)ω/kBT |χn(u)|2 = 1

  • 2πu2T

exp

  • −

u2 2u2T

  • Meheler formula

displacement u energy χ1 χ2 χ3 displacement u Prob (u) T

Large fluctuations of DFT eigenvalues are possible in MD, but statistically not significant

Giustino, Lecture Tue.1 29/31

slide-83
SLIDE 83

Take-home messages

  • We can understand the basics of electron-phonon physics

using elementary perturbation theory

  • The calculations almost invariably require a fine sampling
  • f the matrix elements across the Brillouin zone
  • The electron-phonon coupling constant λ was introduced

to study metals and superconductors

  • Rayleigh-Schr¨
  • dinger perturbation theory and MD

simulations describe the same physics

Giustino, Lecture Tue.1 30/31

slide-84
SLIDE 84

References

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

(North-Holland, Amsterdam)

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

[link]

  • S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys.

73, 515 (2001) [link]

  • N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, D. Vanderbilt, Rev. Mod.
  • Phys. 84, 1419 (2012)

[link]

  • C. E. Patrick and F. Giustino, J. Phys. Condens. Matter 26, 365503

(2014) [Link]

Giustino, Lecture Tue.1 31/31