Excited States Electronic Properties · 2016-06-14 · Excited States Electronic Properties and Theoretical Spectroscopy ... Time-Dependent Density Functional Theory. IntroMBPT &

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

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

En

erg

y(e

V)

Band Structure of SnO2



Z A M Γ X R Z Γ A R Γ

-10

-8

-6

-4

-2

0

2

4

6

8

10

12E

ner

gy

(eV

)

Band Structure of SnO2



calculated

gap(eV)

:LDA

HgTe InSb

,P,InAs

InN,Ge,GaS

b,CdO

SiInP,GaA

s,CdTe,AlSb

Se,Cu2O

AlAs,GaP

,SiC,AlP,CdS

ZnSe

,CuB

rZnO,GaN

,ZnS

diam

ond

SrO AlN

MgO

CaO

experimental gap (eV)


Optical properties

11 12 13 14 15 16ω(eV)

5

10

15

Im

εM

exp

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



i

unoccupied states

occupied states

j

Absorption



Electron Energy Loss Spectroscopy


Outline







DFT vs MBPT


O[n] ⇐ n(r) =occ∑i

|φi (r)|2

Many-Body Perturbation Theory

O[G ]

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

]∣∣∣N⟩


DFT vs MBPT



|φi (r)|2


O[G ]

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

]∣∣∣N⟩


DFT vs MBPT



|φi (r)|2


O[G ]

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

En =

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

En =

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( )

E


Green’s Function G - spectral function

A(r1, r2, ω) ∝ ImG (r1, r2, ω)

A( )

E

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( )

E




[−∇2 + Vion + VH + Σ(r, r′,E )


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

• Ei complexquasiparticle

interacting

non-interactingA( )

EReE i

ImE 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

xc




interacting

non-interactingA( )

EReE i

ImE i

ilda


LDA band gap

calculated

gap(eV)

:LDA

HgTe InSb

,P,InAs

InN,Ge,GaS

b,CdO

SiInP,GaA

s,CdTe,AlSb

Se,Cu2O

AlAs,GaP

,SiC,AlP,CdS

ZnSe

,CuB

rZnO,GaN

,ZnS

diam

ond

SrO AlN

MgO

CaO



GW results :: band gap

calculated

gap(eV)

:LDA:GW(LDA)

HgT

e InSb

,P,In

AsInN,Ge,GaS

b,CdO

SiInP,GaA

s,CdT

e,AlSb

Se,Cu2

OAlAs

,GaP

,SiC,AlP,CdS

ZnSe

,CuB

rZn

O,GaN

,ZnS

diam

ond

SrO AlN

MgO

CaO


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


GW results :: band gap


QPs

cGW

gap(eV)

MgOAlN

CaO

HgT

e InSb

,InAs

InN,GaS

bInP,GaA

s,CdT

eCu2

O ZnTe

,CdS

ZnSe

,CuB

rZn

O,GaN

ZnS

P,Te

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

Ei

1 + ErEi

)


Absorption

Creation of an electron-hole pair

i

unoccupied states

occupied states

j


Absorption

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


Absorption

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


Spectroscopy: Electron Scattering



Energy Loss Function

d2σ

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

Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

εX-ray

R index




Vtot = ε−1Vext

Vtot = Vext + Vind

E = ε−1D


Abs

EELS

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






ε−1 = 1 + vχ






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)

i

unoccupied states

occupied states

j


Spectra within DFT

Loss spectrum of Graphite


Spectra within DFT


Spectra within DFT

11 12 13 14 15 16ω(eV)

5

10

15

Im

εM

exp

DFT level

Solid Argon Absorption spectrum


Outline








Polarizability





Polarizability




χ0 =∑ij


′)

ω − (εi − εj)





Polarizability



χ0 =∑ij


′)

ω − (εi − εj)

i

unoccupied states

occupied states

j


Spectra within DFT



How to go beyond χ0 ?


Hedin’s pentagon


Hedin’s pentagon


Hedin’s pentagon


Hedin’s pentagon


Spectra in MBPT

Spectra in IP picture

IP-RPA

Abs = Im χ0

i

unoccupied states

occupied states

j


Spectra in MBPT

Spectra in GW approximation

GW-RPA

Abs = Im χ0GW

χ0GW = P = −iGG

ioccupied states

j

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 =

∑ij


ω −[

(εi + ∆GW

i )−(εj + ∆GW

j

)]


Spectra in MBPT

Spectra in GW-RPA

χ0 =∑ij


ω − (εi − εj)

⇓

χ0GW =

∑ij


ω −[

(εi + ∆GW

i )−(εj + ∆GW

j

)]


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)

[v +

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

[v +

δΣ

δG

]P



P = GG + GG

[v − δ [GW ]

δG

]P



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)

TDDFT

• 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

http://etsf.polytechnique.fr/sites/default/files/users/matteo/matteo_thesis.pdf

http://etsf.polytechnique.fr/sites/default/files/users/matteo/matteo_thesis.pdf


Outline








Polarizability





Polarizability




χ0 =∑ij


′)

ω − (εi − εj)





Polarizability



χ0 =∑ij


′)

ω − (εi − εj)

i

unoccupied states

occupied states

j



Polarizability



m

Density Functional Formalism

δn = δnn−i

δVtot = δVext + δVH + δVxc



Polarizability

χδVext = χ0 (δVext + δVH + δVxc)

χ = χ0

(1 +

δVH

δVext+δVxc

δVext

)δVH

δVext=δVH

δn

δn

δVext= vχ

δVxc

δVext=δVxc

δn

δn

δVext= fxcχ

with fxc = exchange-correlation kernel



Polarizability


χ = χ0

(1 +

δVH

δVext+δVxc

δVext

)δVH

δVext=δVH

δn

δn

δVext= vχ

δVxc

δVext=δVxc

δn

δn

δVext= fxcχ




Polarizability


χ = χ0

(1 +

δVH

δVext+δVxc

δVext

)δVH

δVext=δVH

δn

δn

δVext= vχ

δVxc

δVext=δVxc

δn

δn

δVext= fxcχ

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



Polarizability


χ = χ0

(1 +

δVH

δVext+δVxc

δVext

)δVH

δVext=δVH

δn

δn

δVext= vχ

δVxc

δVext=δVxc

δn

δn

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0




Polarizability


χ = χ0

(1 +

δVH

δVext+δVxc

δVext

)δVH

δVext=δVH

δn

δn

δVext= vχ

δVxc

δVext=δVxc

δn

δn

δVext= fxcχ

χ =[1− χ0 (v + fxc)

]−1χ0




Polarizability χ in TDDFT

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

2. φi , εi → χ0 =∑

ij

φi (r)φ∗j (r)φ∗i (r′)φj (r

′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc

variation of the potentials

4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc

δn“any” other function





2. φi , εi → χ0 =∑

ij


′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc


4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc






2. φi , εi → χ0 =∑

ij


′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc


4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc






2. φi , εi → χ0 =∑

ij


′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc


4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc






2. φi , εi → χ0 =∑

ij


′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc


4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc






2. φi , εi → χ0 =∑

ij


′)

ω−(εi−εj )

3.δVH

δn= v

δVxc

δn= fxc


4. χ = χ0 + χ0 (v + fxc)χ

A comment

• fxc =

δVxc




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

Why?

f ALDAxc is short-range

fxc(q→ 0) ∼ 1

q2



Good results


• ELS of solids

Bad results


Why?


fxc(q→ 0) ∼ 1

q2



Good results


• ELS of solids

Bad results


Why?


fxc(q→ 0) ∼ 1

q2



Absorption of Silicon fxc = αq2

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

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Excited States Electronic Properties · 2016-06-14 · Excited States Electronic Properties and Theoretical Spectroscopy ... Time-Dependent Density Functional Theory. IntroMBPT &