excited states electronic properties · 2016-06-14 · excited states electronic properties and...

Intro MBPT & GW Linear BSE TDDFT

Excited States Electronic Propertiesand Theoretical Spectroscopy

Francesco Sottile

Ecole de simulation numerique en matiere condensee

Jussieu, 9 June 2016

Outline

Introduction and reminder of ground-state results

Photoemission via MBPT: GW approximation

Absorption and Loss spectroscopies: linear response quantities

Neutral excitations within MBPT: Bethe-Salpeter Equation

Neutral excitations within DFT: Time-Dependent DensityFunctional Theory

Outline

Density Functional Theory

E [n] ⇒

Total energy, phase stability,bulk modulus, lattice constant, etc.[

−∇2 + Vion + VH + Vxc

]φi (r) = εiφi (r)

n(r) =∑i

|φi (r)|2

E [n] ⇒

n(r) =∑i

|φi (r)|2

E [n] ⇒

n(r) =∑i

|φi (r)|2

E [n] ⇒

n(r) =∑i

|φi (r)|2

Often reasonable results also for excitations

• ab initio

• qualitative estimate

• powerful analysis tool

• starting point for more accurate methods

homo lumo

Z A M Γ X R Z Γ A R Γ

Band Structure of SnO2

Z A M Γ X R Z Γ A R Γ

Band Structure of SnO2

calculated

gap(eV)

HgTe InSb

,P,InAs

InN,Ge,GaS

SiInP,GaA

s,CdTe,AlSb

Se,Cu2O

AlAs,GaP

,SiC,AlP,CdS

rZnO,GaN

SrO AlN

experimental gap (eV)

Optical properties

11 12 13 14 15 16ω(eV)

DFT level

Solid Argon Absorption spectrum

E [n] ⇒

n(r) =∑i

|φi (r)|2

What is an electron ?

Spectroscopies :: charged excitations

Direct Photoemission

Inverse Photoemission

Direct and inverse photoemission spectroscopy

Spectroscopies :: neutral excitations

Absorption

unoccupied states

occupied states

Absorption

Electron Energy Loss Spectroscopy

Outline

DFT vs MBPT

O[n] ⇐ n(r) =occ∑i

|φi (r)|2

Many-Body Perturbation Theory

G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)

]∣∣∣N⟩

DFT vs MBPT

|φi (r)|2

G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)

]∣∣∣N⟩

DFT vs MBPT

|φi (r)|2

G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)

]∣∣∣N⟩

Green’s Function G

G (r1, r2, ω) =∑n

An(r1)A∗n(r2)

ω − En + iη

additional energy :: En > µ

removal energy :: En < µ

The poles of G are the excitation energies

Green’s Function G

G (r1, r2, ω) =∑n

An(r1)A∗n(r2)

ω − En + iη

additional energy :: En > µ

removal energy :: En < µ

The poles of G are the excitation energies

Direct Photoemission

Green’s Function G - spectral function

A(r1, r2, ω) ∝ ImG (r1, r2, ω)

interacting

non-interactingA( )

Green’s Function G - spectral function

A(r1, r2, ω) ∝ ImG (r1, r2, ω)

non-interacting

interacting

How to calculate G ?

G (1, 2) = −i⟨N∣∣∣T [ψ(1)ψ†(2)

]∣∣∣N⟩G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Σ = self-energy

How to calculate G ?

G (1, 2) = −i⟨N∣∣∣T [ψ(1)ψ†(2)

]∣∣∣N⟩G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Σ = self-energy

Hedin’s equations

Σ(1, 2) = i

∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)

G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

W (1, 2) = v(1, 2) +

∫d(34)v(1, 3)P(3, 4)W (4, 2)

Hedin’s equations

Σ(1, 2) = i

∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)

G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

W (1, 2) = v(1, 2) +

∫d(34)v(1, 3)P(3, 4)W (4, 2)

Hedin’s equations

Σ(1, 2) = i

∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)

G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

W (1, 2) = v(1, 2) +

∫d(34)v(1, 3)P(3, 4)W (4, 2)

Hedin’s equations

Σ(1, 2) = i

∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)

G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

W (1, 2) = v(1, 2) +

∫d(34)v(1, 3)P(3, 4)W (4, 2)

Hedin’s equations

Σ(1, 2) = i

∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)

G (1, 2) = G 0(1, 2) +

∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

W (1, 2) = v(1, 2) +

∫d(34)v(1, 3)P(3, 4)W (4, 2)

Hedin’s pentagon

Hedin’s pentagon: possible strategy

In practice: quasi-particle approximation

Quasi-particle equation

[−∇2 + Vion + VH + Σ(r, r′,E )

]ψi (r) = Eiψi (r)

[−∇2 + Vion + VH + Σ(r, r′,E )

interacting

non-interactingA( )

[−∇2 + Vion + VH + Σ(r, r′,E )

• Σ(r, r′, ω) non-local,non-hermitian, energydependent

• Ei complexquasiparticle

interacting

non-interactingA( )

EReE i

[−∇2 + Vion + VH + Σ(r, r′,E )

DFT-KS equation

[−∇2 + Vion + VH + Vxc

[−∇2 + Vion + VH + G 0W 0

DFT-KS equation

[−∇2 + Vion + VH + V lda

interacting

non-interactingA( )

EReE i

LDA band gap

calculated

gap(eV)

HgTe InSb

,P,InAs

InN,Ge,GaS

SiInP,GaA

s,CdTe,AlSb

Se,Cu2O

AlAs,GaP

,SiC,AlP,CdS

rZnO,GaN

SrO AlN

GW results :: band gap

calculated

gap(eV)

:LDA:GW(LDA)

e InSb

AsInN,Ge,GaS

SiInP,GaA

e,AlSb

Se,Cu2

,SiC,AlP,CdS

SrO AlN

van Schilfgaarde et al., PRL 96, 226402 (2006)

GW results :: band gap

gap(eV)

MgOAlN

e InSb

InN,GaS

bInP,GaA

O ZnTe

SiGe,CdO

AlSb,SeAlAs,GaP,SiC,AlP

SrOdiamond

van Schilfgaarde et al., PRL 96, 226402 (2006)

Outline

Absorption

Beer Law

I (x) = I0e−αx

α⇐⇒ ε

Absorption

Ellipsometry Experiments

ε = sin2Φ + sin2Φtan2Φ

(1− Er

1 + ErEi

Absorption

Creation of an electron-hole pair

unoccupied states

occupied states

Absorption

Lautenschlager et al., PRB 36, 4821 (1987)

Absorption

Izumi et al., Anal.Chem. 77, 6969 (2005)

Spectroscopy: Electron Scattering

Energy Loss Function

dΩdE∝ Im

ε−1

Spectroscopy: X-ray Scattering

Linear Response Approach

System submitted to an external perturbation

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

Dielectric function ε

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D

εX-ray

R index

Definition of polarizability

not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1

ε−1 = 1 + vχ

χ is the polarizability of the system

ε−1 = 1 + vχ

Polarizability

interacting system δn = χδVext

non-interacting system δnn−i = χ0δVtot

Polarizability

Single-particle polarizability

χ0 =∑ij

φi (r)φ∗j (r)φ∗i (r′)φj(r

ω − (εi − εj)

hartree, hartree-fock, dft, etc.

G.D. Mahan Many Particle Physics (Plenum, New York, 1990)

Polarizability

χ0 =∑ij

ω − (εi − εj)

unoccupied states

occupied states

Spectra within DFT

Loss spectrum of Graphite

Spectra within DFT

11 12 13 14 15 16ω(eV)

DFT level

Solid Argon Absorption spectrum

Outline

Polarizability

χ0 =∑ij

ω − (εi − εj)

Polarizability

χ0 =∑ij

ω − (εi − εj)

unoccupied states

occupied states

Spectra within DFT

How to go beyond χ0 ?

Hedin’s pentagon

Spectra in MBPT

Spectra in IP picture

IP-RPA

Abs = Im χ0

unoccupied states

occupied states

Spectra in MBPT

Spectra in GW approximation

GW-RPA

Abs = Im χ0GW

χ0GW = P = −iGG

ioccupied states

unoccupied (GW corrected) states

Spectra in MBPT

Spectra in GW-RPA

χ0 =∑ij

φi(r)φ∗j (r)φ∗i (r′)φj(r′)

ω − (εi − εj)

χ0GW =

ω −[

(εi + ∆GW

i )−(εj + ∆GW

Spectra in MBPT

Spectra in GW-RPA

χ0 =∑ij

ω − (εi − εj)

χ0GW =

ω −[

(εi + ∆GW

i )−(εj + ∆GW

Spectra in MBPT

Spectra in GW-RPA

Spectra in MBPT

GG Polarizability

P(1, 2) = −i G (1, 2)G (2, 1+)

Spectra in MBPT

GGΓ Polarizability

P(1, 2) = −i∫

d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)

Bethe-Salpeter Equation

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+

∫d(4567)

δΣ(1, 2)

δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)

Towards the Bethe-Salpeter Equation

From electron and hole propagation .....

P0(1234) = G (13)G (42) ...

Towards the Bethe-Salpeter Equation

From electron and hole propagation to the electron-holeinteraction

P(1234) = P0(1234) + P0(1256)

δΣ(56)

δG (78)

]P(7834)

P(1234)=P0(1234)+P0(1256)

[v(57)δ(56)δ(78)+

δΣ(56)

δG (78)

]P(7834)

P = P0 + P0

P = GG + GG

[v − δ [GW ]

P = GG + GG [v −W ]P

Bethe-Salpeter EquationBethe-Salpeter Equation

P = P0 + P0 [v −W ]P

Intrinsic 4-point equation

Correct!It describes the (coupled) progation

of two particles, the electron andthe hole !

P(1234) = P0(1234)+

+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)

P(1234) = P0(1234)+

+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)

P(1234) = P0(1234)+

+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)

Bethe-Salpeter equation results: Semiconductors

Albrecht et al., PRL 80, 4510 (1998)

Bethe-Salpeter equation results: Insulators

Sottile et al., PRB 76, 161103 (2007).

Bethe-Salpeter equation results: Molecule (Na4)

Onida et al., PRL 75, 818 (1995)

Bethe-Salpeter equation results: Silicon Nanowires

Bruno et al., PRL 98, 036807 (2007)

Bethe-Salpeter equation results: Hexagonal Ice

Hahn et al., PRL 94, 37404 (2005)

Bethe-Salpeter equation results: EELS of Silicon

Olevano and Reining, PRL 86, 5962 (2001)

Bethe-Salpeter equation results: Surface

Rohlfing et al., PRL 85, 005440 (2000)

Bethe-Salpeter equation results: Surface

Rohlfing et al., PRL 85, 005440 (2000)

Bethe-Salpeter equation: State-of-the-art

• DFT - ground state

• GW - quasiparticle energies

• BSE - optical and dielectric properties

√several spectroscopies

√variety of systems

× Cumbersome Calculations

Bethe-Salpeter equation: State-of-the-art

• DFT - ground state

• GW - quasiparticle energies

• BSE - optical and dielectric properties

√several spectroscopies

√variety of systems

× Cumbersome Calculations

References and Literature

GW and BSE

• Hedin, Lundqvist, Solid State Physics 23, 1 (1969)

• Onida, Reining, Rubio, RMP 74, 601 (2002)

• Strinati, Riv Nuovo Cimento 11, 1 (1988)

• Runge, Gross, Kohn, PRL 52, 997 (1984), PRL 55, 2850(1985)

• Marques et al eds, Time Dependent Density FunctionalTheory, Springer (2006).

• Botti et al, Rep. Prog. Phys. 70, 357 (2007)

Matteo Gatti, PhD Thesis, http://etsf.polytechnique.fr/sites/default/files/users/matteo/matteo_thesis.pdf

Outline

Polarizability

χ0 =∑ij

ω − (εi − εj)

Polarizability

χ0 =∑ij

ω − (εi − εj)

unoccupied states

occupied states

Polarizability

Density Functional Formalism

δn = δnn−i

δVtot = δVext + δVH + δVxc

Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

δVext+δVxc

δVext

δVext=δVH

δVext= vχ

δVext=δVxc

δVext= fxcχ

with fxc = exchange-correlation kernel

Polarizability

χ = χ0

δVext+δVxc

δVext

δVext=δVH

δVext= vχ

δVext=δVxc

δVext= fxcχ

Polarizability

χ = χ0

δVext+δVxc

δVext

δVext=δVH

δVext= vχ

δVext=δVxc

δVext= fxcχ

χ = χ0 + χ0 (v + fxc)χwith fxc = exchange-correlation kernel

Polarizability

χ = χ0

δVext+δVxc

δVext

δVext=δVH

δVext= vχ

δVext=δVxc

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0

Polarizability

χ = χ0

δVext+δVxc

δVext

δVext=δVH

δVext= vχ

δVext=δVxc

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0

Polarizability χ in TDDFT

1. DFT ground-state calc. → φi , εi [Vxc ]

2. φi , εi → χ0 =∑

φi (r)φ∗j (r)φ∗i (r′)φj (r

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

variation of the potentials

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δn“any” other function

2. φi , εi → χ0 =∑

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

2. φi , εi → χ0 =∑

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

2. φi , εi → χ0 =∑

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

2. φi , εi → χ0 =∑

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

2. φi , εi → χ0 =∑

ω−(εi−εj )

3.δVH

δn= v

δn= fxc

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

Most important approximation for fxc

fxc = 0 RPA

f ALDAxc (r, r′) = δVxc (r)

δn(r′) δ(r − r′) ALDA

Role of v

χ(r, r′, ω) = χ0(r, r′, ω) + χ0(r, r′′, ω)v(r′′, r′′′)χ(r′′′, r′, ω)

⟨χ(r, r′, ω)

⟩⇒⟨χ0(r, r′, ω)

⟩spectrum

(χ(|r − r′|, ω)

)spectrum

(χ(r, r′, ω)

)v contains all the information of the crystal localfield effects (local dishomogeneity of the system)

Role of v

χ(r, r′, ω) = χ0(r, r′, ω)

⟨χ(r, r′, ω)

⟩⇒⟨χ0(r, r′, ω)

⟩spectrum

(χ(|r − r′|, ω)

)spectrum

(χ(r, r′, ω)

Role of v

χ(r, r′, ω) = χ0(r, r′, ω)

⟨χ(r, r′, ω)

⟩⇒⟨χ0(r, r′, ω)

⟩spectrum

(χ(|r − r′|, ω)

)spectrum

(χ(r, r′, ω)

Role of v

χ(r, r′, ω) = χ0(r, r′, ω) + χ0(r, r′′, ω)v(r′′, r′′′)χ(r′′′, r′, ω)

⟨χ(r, r′, ω)

⟩⇒⟨χ0(r, r′, ω)

⟩spectrum

(χ(|r − r′|, ω)

)spectrum

(χ(r, r′, ω)

Spectra within DFT and TDDFT-RPA

Loss spectrum of Graphite

A.Marinopoulos et al. Phys.Rev.Lett 89, 76402 (2002)

ALDA: Achievements and Shortcomings

Inelastic X-ray scattering of Silicon

H-C.Weissker et al., Physical Review Letters 97, 237602 (2006)

Photo-absorption cross section of Benzene

K.Yabana and G.F.Bertsch Int.J.Mod.Phys.75, 55 (1999)

Absorption Spectrum of Silicon

Absorption Spectrum of Argon

Good results

• Photo-absorption ofsmall molecules

• ELS of solids

Bad results

• Absorption of solids

f ALDAxc is short-range

fxc(q→ 0) ∼ 1

Good results

• ELS of solids

Bad results

fxc(q→ 0) ∼ 1

Good results

• ELS of solids

Bad results

fxc(q→ 0) ∼ 1

Absorption of Silicon fxc = αq2

L.Reining et al. Phys.Rev.Lett. 88, 66404 (2002)

excited states electronic properties · 2016-06-14 · excited states electronic properties and...

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