Introduction to GW and Bethe-Salpeter - TDDFT.org · Introduction to GW and Bethe-Salpeter beyond density functional theory for electronic excitations Silvana Botti ... Silvana Botti

Introduction to GW and Bethe-Salpeterbeyond density functional theory for electronic excitations

Silvana Botti

1LSI, Ecole Polytechnique-CNRS-CEA, Palaiseau, France2LPMCN, CNRS-Universite Lyon 1, France

3European Theoretical Spectroscopy Facility

June 18, 2010 – Lyon

Silvana Botti Intro to GW and BSE 1 / 45

Outline

1 Starting point: density functional theory

2 Photoemission and optical absorption

3 Beyond ground-state density functional theory

4 Green’s functions and GW approximation

5 Bethe-Salpeter equation and excitons

6 Summary


Starting point: density functional theory

Outline






6 Summary



Modeling electronic excitations in complex systems

ObjectivesPredict accurate values forfundamental opto-electronicalproperties (gap, absorptionspectra, excitons, . . .)

Simulate real materials(nanostructured systems,large unit cells, defects,doping, interfaces, . . .)



How do we study electronic excitations?

In a quantum-mechanical framework that describes, fromfirst-principles, a system of interacting electrons and nuclei in presenceof time-dependent external fieldsMany different possible ways:

Wave-function based: Hartree-Fock and Post-Hartree-Fockmethods, Monte Carlo, . . .Green’s function based: Many-body Perturbation Theory (GW,BSE)Density based: TDDFT

In order to study complex systems it is important to find a compromisebetween accuracy of results and computational effort















The heart of density functional theory (DFT)

There is the 1-to-1 mapping between different external potentials v(r)and their corresponding ground state densities ρ(r):

(i) all observable quantities of a quantum system are completelydetermined by the density

(ii) which means that the basic variable is no more the many-bodywavefunction Ψ (r) but the electron density ρ(r)

Hohenberg and Kohn, Phys. Rev. 136, B864 (1964)



The heart of density functional theory (DFT)

There is the 1-to-1 mapping between different external potentials v(r)and their corresponding ground state densities ρ(r):

(i) all observable quantities of a quantum system are completelydetermined by the density

(ii) which means that the basic variable is no more the many-bodywavefunction Ψ (r) but the electron density ρ(r)

Hohenberg and Kohn, Phys. Rev. 136, B864 (1964)



Band structures from density functional theory

Single-particle Kohn-Sham (KS) equations[−∇

2

2+ vext (r) + vHartree (r) + vxc (r)

]ϕKS

i (r) = εKSi ϕKS

i (r)

ρ (r) =occ.∑

i

|ϕKSi (r) |2

In the KS scheme it is necessary to approximate vxc (r)

It is common to interpret the solutions of the Kohn-Shamequations as one-electron states

Kohn and Sham, Phys. Rev. 140, A1133 (1965)





2


]ϕKS

i (r) = εKSi ϕKS

i (r)

ρ (r) =occ.∑

i

|ϕKSi (r) |2








2


]ϕKS

i (r) = εKSi ϕKS

i (r)

ρ (r) =occ.∑

i

|ϕKSi (r) |2






Density functional theory

Standard computational approach for band structures:

Kohn-Sham (KS) equations[−∇

2


]ϕi (r) = εiϕi (r)

it is necessary to approximate vxc (r),Structural parameters and formation energies are usually good inLDA or GGAKohn-Sham energies are not meant to reproduce quasiparticleband structures: one often obtains good band dispersions butband gaps are systematically underestimatedHow to calculate the optical absorption?

Hohenberg&Kohn, PR 136, B864 (1964); Kohn&Sham, 140, A1133 (1965)



Density functional theory

Time-dependent extension of DFT to access neutral excitations:

Time-dependent KS equations

−i∂

∂tϕi (r , t) =

[−∇

2

2+ vext(r , t) + vHartree[n](r , t) + vxc[n](r , t)

]ϕi (r , t)

it is necessary to approximate vxc (r , t), or its functional derivative,the xc kernel fxc within linear responseSpectra obtained with standard (adiabatic) functionals are goodfor finite systemsHowever, absorption spectra for extended systems are bad

Runge and Gross, Phys. Rev. Lett. 52, 997 (1984)


Photoemission and optical absorption

Outline






6 Summary



Excitation energies: photoemission

Photoemission process:

hν − (Ekin + φ) = EN−1,v − EN,0 = −εv



Excitation energies: photoemission

Inverse photoemission process:

hν − (Ekin + φ) = EN,0 − EN+1,c = −εc



Excitation energies: energy gap

Photoemission gap:Egap = I − A = mink ,l

(EN−1,k + EN+1,l − 2EN,0

)Silvana Botti Intro to GW and BSE 11 / 45


Excitation energies: energy gap

Optical gap:Egap = I − A− Eexc

binding



An intuitive Picture: Absorption

v

c

unoccupied states

occupied states

Independent particle KS picture using εKSi and ϕKS

i



An intuitive Picture: Absorption

v

c

unoccupied states

occupied states

Electron-hole interaction: Excitons!



The simplest way: independent-particle transitions

v

c

unoccupied states

occupied states

Fermi’s golden rule: χKS ∼∑

v ,c |〈c|D|v〉|2 δ(εc − εv − ω)



The simplest way

v

c

unoccupied states

occupied states

ε2(ω) = 24π2

ΩNkω2 limq→0

1q2

∑v ,c,k

∣∣mv ,c,k∣∣2 δ(εck − εvk − ω)

mv ,c,k = 〈c |q · v| v〉 velocity matrix elements



Joint density of states

In the independent-particle approximation the dielectric function isdetermined by two contributions: optical matrix elements and energylevels.

ε2(ω) = 24π2

ΩNkω2 limq→0

1q2

∑v ,c,k

∣∣mv ,c,k∣∣2 δ(εck − εvk − ω)

If mv ,c,k can be considered constant then the spectrum is essentiallygiven by the joint density of states:

ε2 ∝ JDOS/ω2 =1

Nkω2

∑v ,c,k

δ(εck − εvk − ω)


Beyond ground-state density functional theory

Outline






6 Summary



Time-dependent DFT for neutral excitations

Runge-Gross theoremThere is a one-to-one correspondence between the time-dependentdensity and the external potential, ρ(r , t)↔ v(r , t)

The many-body equation is mapped onto the time-dependentKohn-Sham equation:

−i∂

∂tφi(r , t) =

[−∇

2


]φi(r , t)

it is necessary to approximate vxc (r , t), or its functional derivative,the xc kernel fxc within linear response




Time-dependent DFT for neutral excitations

Runge-Gross theoremThere is a one-to-one correspondence between the time-dependentdensity and the external potential, ρ(r , t)↔ v(r , t)

The many-body equation is mapped onto the time-dependentKohn-Sham equation:

−i∂

∂tφi(r , t) =

[−∇

2


]φi(r , t)

it is necessary to approximate vxc (r , t), or its functional derivative,the xc kernel fxc within linear response




Beyond independent-particle picture – the TDDFT way

Within linear response, χ(r, r′, ω) is the reducible polarizability

V

Vind

ext

macroscopic and microscopic

macroscopic

↑δn = χvext

χ = χKS + χKS (v + fxc)χ

v = Coulomb potential, related to local field effectsfxc = quantum exchange-correlation effects



Linear response functions

χ = χKS + χKS

(v︸︷︷︸+ fxc︸︷︷︸

)χ

classical local field effects exchange-correlation effects

Independent-particle: v = 0, fxc = 0RPA: fxc = 0TDLDA: fxc in the adiabatic LDABSE-based kernels for solids



Linear response functions

χ = χKS + χKS

(v︸︷︷︸+ fxc︸︷︷︸

)χ

classical local field effects exchange-correlation effects

Independent-particle: v = 0, fxc = 0RPA: fxc = 0TDLDA: fxc in the adiabatic LDABSE-based kernels for solids


Green’s functions and GW approximation

Outline






6 Summary



A more “intuitive” path: electrons and holes

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In the many-body Green’s function framework weseparate two processes:

GW for electron addition and removal(one-particle G)Bethe-Salpeter equation for the inclusion ofelectron-hole interaction (two-particle G)

The price to pay is a more involved theoretical andcomputational framework

L. Hedin, Phys. Rev. 139 (1965)



Green’s functions

Green’s function: propagation of an extra-particle

G(r1, r2, t1 − t2) = −i〈N|T [ψ(r1, t1)ψ†(r2, t2)]|N〉

Electron density:

ρ (r) = G(r , r , t , t+)

Spectral function:

A(ω) = 1/πTr Im G(r1, r2, ω)



Hedin’s equations

Σ

G

ΓP

W

G=G 0+G 0 Σ G

Γ=1+

(δΣ/

δG)G

GΓ

P = GGΓ

W = v + vPW

Σ = GWΓ

L. Hedin, Phys. Rev. 139 (1965).



Self-energy and screened interaction

Self-energy: nonlocal, non-Hermitian, frequency dependent operatorIt allows to obtain the Green’s function G once that G0 is known

Hartree-Fock Σx (r1, r2) = iG(r1, r2, t , t+)v(r1, r2)

GW Σ(r1, r2, t1 − t2) = iG(r1, r2, t1 − t2)W (r1, r2, t2 − t1)

W = ε−1v : screened potential (much weaker than v !)

Ingredients:KS Green’s function G0, and RPA dielectric matrix ε−1

G,G′(q, ω)

L. Hedin, Phys. Rev. 139 (1965)



Perturbative GW: “best G, best W”

Kohn-Sham equation:

H0(r)ϕKS (r) + vxc (r)ϕKS (r) = εKSϕKS (r)

Quasiparticle equation:

H0(r)φQP (r) +

∫dr ′Σ

(r , r ′, ω = EQP

)φQP

(r ′)

= EQPφQP (r)

Quasiparticle energies 1st order perturbative correction with Σ = iGW :

EQP − εKS = 〈ϕKS|Σ− vxc|ϕKS〉

Basic assumption: φQP ' ϕKS

Hybersten&Louie, PRB 34 (1986); Godby, Schluter&Sham, PRB 37 (1988)



Perturbative GW: “best G, best W”

Kohn-Sham equation:

H0(r)ϕKS (r) + vxc (r)ϕKS (r) = εKSϕKS (r)

Quasiparticle equation:

H0(r)φQP (r) +

∫dr ′Σ

(r , r ′, ω = EQP

)φQP

(r ′)

= EQPφQP (r)

Quasiparticle energies 1st order perturbative correction with Σ = iGW :

EQP − εKS = 〈ϕKS|Σ− vxc|ϕKS〉

Basic assumption: φQP ' ϕKS

Hybersten&Louie, PRB 34 (1986); Godby, Schluter&Sham, PRB 37 (1988)



LDA Kohn-Sham and G0W0 energy gaps

0

2

4

6

8

calc

ula

ted g

ap (

eV

)

:LDA

:GW(LDA)

HgT

e

InS

b,P

,InA

sIn

N,G

e,G

aS

b,C

dO

Si

InP

,GaA

s,C

dT

e,A

lSb

Se,C

u2O

AlA

s,G

aP

,SiC

,AlP

,CdS

ZnS

e,C

uB

r

ZnO

,GaN

,ZnS

dia

mond

SrO A

lN

MgO

CaO

van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006)


experimental gap (eV)


LDA and G0W0 energies for CIS

CuInS2DFT-LDA G0W0 exp.

Eg -0.11 0.28 1.54In-S 6.5 6.9 6.9

S s band 12.4 13.0 12.0In 4 d band 14.6 16.4 18.2

CuInSe2DFT-LDA G0W0 exp.

Eg -0.29 0.25 1.05In-Se 5.8 6.15 6.5

Se s band 12.6 12.9 13.0In 4 d band 14.7 16.2 18.0

www.abinit.org


www.abinit.org


LDA and G0W0 energies for CIS

CuInS2DFT-LDA G0W0 exp.

Eg -0.11 0.28 1.54In-S 6.5 6.9 6.9

S s band 12.4 13.0 12.0In 4 d band 14.6 16.4 18.2

CuInSe2DFT-LDA G0W0 exp.

Eg -0.29 0.25 1.05In-Se 5.8 6.15 6.5

Se s band 12.6 12.9 13.0In 4 d band 14.7 16.2 18.0

www.abinit.org


www.abinit.org


Beyond Standard GW

Looking for another starting point:DFT with another approximation for vxc : GGA, EXX,...(e.g. Rinke et al. 2005)LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al. 2009 )Hybrid functionals (e.g. Fuchs et al. 2007)

Self-consistent approaches:GWscQP scheme (Faleev et al. 2004)scCOHSEX scheme (Hedin 1965, Bruneval et al. 2005)

Our choice is to get a better starting point for G0W0 using scCOHSEX



Beyond Standard GW

Looking for another starting point:DFT with another approximation for vxc : GGA, EXX,...(e.g. Rinke et al. 2005)LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al. 2009 )Hybrid functionals (e.g. Fuchs et al. 2007)

Self-consistent approaches:GWscQP scheme (Faleev et al. 2004)scCOHSEX scheme (Hedin 1965, Bruneval et al. 2005)

Our choice is to get a better starting point for G0W0 using scCOHSEX



Hedin’s equations

Σ

G

ΓP

W

G=G 0+G 0 Σ G

Γ=1+

(δΣ/

δG)G

GΓ

P = GGΓ

W = v + vPW

Σ = GWΓ




Hedin’s equations

Σ

G

ΓP

W

G=G 0+G 0 Σ G

Γ=1+

(δΣ/

δG)G

GΓ

P = GGΓ

W = v + vPW

Σ = GWΓ

P = GG




COHSEX: approximation to GW self-energy

Coulomb hole:

ΣCOH(r1, r2) =12δ(r1 − r2)[W (r1, r2, ω = 0)− v(r1, r2)]

Screened Exchange:

ΣSEX(r1, r2) = −∑

i

θ(µ− Ei)φi(r1)φ∗i (r2)W (r1, r2, ω = 0)

The COHSEX self-energy is static and HermitianSelf-consistency can be done either on energies alone or on bothenergies and wavefunctionsRepresentation of WFs on a restricted LDA basis set



Self-consistent GW a la Faleev

Make self-energy Hermitian and static

〈k i |Σ|k j〉 =14(〈k i |Σ(εk j)|k j〉+ 〈k j |Σ(εk j)|k i〉∗

+〈k i |Σ(εk i)|k j〉+ 〈k j |Σ(εk i)|k i〉∗)

|k i〉 and εk i are self-consistent eigensolutions of the iterativeprocedureRepresentation of WFs on a restricted LDA basis setRequires sums over empty states

Faleev, van Schilfgaarde, and Kotani, PRL 93 2004



Self-consistent COHSEX

In both sc approaches the self-energy is made hermitian and static

Advantages of sc-COHSEXphysically motivated: accounts for Coulomb-hole andscreened-exchangecomputationally friendly: only occupied statessc-COHSEX wave-functions very similar to GWscQP onesstill a “best G, best W” approachdynamical correlations added in the G0W0 step

Bruneval et al. PRL 97, 267601 (2006), Gatti et al. PRL 99, 266402 (2007)



Energy gap within sc-GW

0 2 4 6 8

0

2

4

6

8

experimental gap (eV)

QP

scG

W g

ap

(e

V)

MgO

AlN

CaO

HgT

e InS

b,I

nA

sIn

N,G

aS

b

InP

,Ga

As,C

dT

eC

u2

O Zn

Te

,Cd

SZ

nS

e,C

uB

rZ

nO

,Ga

NZ

nS

P,Te

SiGe,CdO

AlSb,SeAlAs,GaP,SiC,AlP

SrOdiamond

van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006)



Quasiparticle energies within sc-GW for CIS

CuInS2DFT-LDA G0W0 sc-GW exp.

Eg -0.11 0.28 1.48 1.54In-S 6.5 6.9 7.0 6.9

S s band 12.4 13.0 13.6 12.0In 4 d band 14.6 16.4 18.2 18.2

CuInSe2DFT-LDA G0W0 sc-GW exp.

Eg -0.29 0.25 1.14 1.05 (+0.2)In-Se 5.8 6.15 6.64 6.5

Se s band 12.6 12.9 13.6 13.0In 4 d band 14.7 16.2 17.8 18.0

sc-GW is here sc-COHSEX+G0W0

www.abinit.org


www.abinit.org

Bethe-Salpeter equation and excitons

Outline






6 Summary



Bethe-Salpeter equation: electron-hole interaction

BSE uses the intuitive quasiparticle picture: it is easier to identifyapproximations.Optical absorption experiment creates an interacting electron-holepair, the exciton.Good agreement between theory and experiment can only beachieved taking into account the exciton, especially if the systemis a semiconductor or an insulator.Small-gap semiconductors and metals screen excitons.The intrinsic two-particle nature of the BSE makes thecalculations very cumbersome, since a four-point equation (due tothe propagation of two particles) has to be solved.

Salpeter and Bethe, Phys. Rev. 84, 1232 (1951)




The four-point reducible polarizability is the key-quantity:

L(1,2,3,4) = L0(1,2,3,4)−G2(1,2,3,4)

The two-particle G2 describes the propagation of two particles, L0is the disconnected part consisting of two one-particle G:

L0 = iG(1,3)G(4,2)

L satisfies a Dyson-like equation, the BSE:

L(1,2,3,4) = L0(1,2,3,4) +

∫d5678 L0(1,2,5,6)

[v(5,7)δ(5,6)δ(7,8) +Ξ(5,6,7,8)] L(7,8,3,4)




The standard approximation for Ξ = iδΣ(5,6)/δG(7,8) is usingthe GW self-energy and approximate to first order in W :

L = L0 + L0

(4v −4 W

)L ,

where 4v(1,2,3,4) = δ(1,2)δ(3,4)v(1,3) and4W = δ(1,3)δ(2,4)W (1,2)

The BSE corresponds to the inclusion of vertex corrections in Pthrough a second iteration of Hedin’s equations.Because of the way the indices are connected for 4W , the BSEcan not be written in a two-point form.The measurable χ is obtained via a two-point contraction of L

χred(1,2) = −L(1,1,2,2)

.Silvana Botti Intro to GW and BSE 39 / 45



In practice, the BSE can be solved by diagonalizing a two-particleexcitonic Hamiltonian which moreover provides information about theexcitonic eigenstates and eigenvalues. In transition space and usingthe only-resonant approximation:

H2p,exc(vc)(v ′c′)A

v ′c′

λ = Eexcλ Av ′c′

λ

The ingredients are:

Kohn-Sham wavefunctions and energiesGW corrected energiesscreening matrix ε−1

GG′(q)



An example: Excitons in CdSe nanowires (NWs)

J. Mater. Chem. 16, 3893 (2006)

CdSe wurtzite nanowires

J.G. Vilhena, S. Botti, and M.A.L. Marques, Appl. Phys. Lett. 96, 123106 (2010)



Importance of classical local fields in 1D

Suppression of the absorption for light polarized ⊥ to the NW axis

http://www.yambo-code.org/




Excitonic effects

Strongly bound excitons for small NW diametersModel TDDFT kernels (LRC curve) valid for bulk CdSe cannot beused for small wires




Summary

Outline






6 Summary


Summary

Summary

Methods that go beyond ground-state DFT are by now well established

However one should remember that interpretation of experiments isoften not straightforwardA better starting point is absolutely necessary for d-electrons

Self-consistent COHSEX+G0W0 gives a very good description ofquasi-particle statesHybrid functionals can be a good compromiseLDA+U can not work when there is hybridization of p − d states

For accurate absorpion spectra when excitonic effects are importantone has to solve the Bethe-Salpeter equation.


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Introduction to GW and Bethe-Salpeter - TDDFT.org · Introduction to GW and Bethe-Salpeter beyond density functional theory for electronic excitations Silvana Botti ... Silvana Botti