Green’s function theory for solid state

electronic band structure

Hong JiangCollege of Chemistry, Peking University

Outline

Introduction: what is Green’s functions for?

Green’s function for single-electron Schrödinger Equations

Green’s function for many-body systems: general formalism

GW approximation and implementation

Applications of the GW approach: examples

Further ReadingsThe GW approach

G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).

W. G . Aulbur, L. Jonsson, J. Wilken, Solid State Phys., 54, 1 (2000).

F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998)

L. Hedin and S. Lundqvist, Solid State Phys. 23, 1-181 (1970).

Many-body theory in general

J. C. Inkson, Many-Body Theory of Solids: An Introduction (Plenum Press, 1984).

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover, 2003).

E. K. U. Gross, E. Runge, O. Heinonen, Many-Particle Theory, (IOP Publishing, 1991).

R. D. Mattuck, A guide to Feynman diagrams in the many-body problem, (McGrow Hill, 1976).

E. N. Economou, Green’s functions in Quantum Physics (3rd ed., Springer, 2006).

Introduction: What are Green’s functions for?

Why are electronic band structure important?

R. M. Navarro Yerga et al. (2009)

Graetzel, Nature (2001)

Electron excitations: experimental measurements

[Yu&Cardona] Yu and Cardona, Fundamentals of Semiconductors(3rd), Springer (2003)

Sample

Source

Source

Analyzer

Analyzer

photon 0: absorption, reflection

photon photon : Raman scattering, Compton scattering, XES

photon electron : PES (XPS, UPS)

electron electron: electron energy loss spectroscopy

electron photon: inverse PES (BIS)

Electron excitations: examples

[Yu&Cardona]

Electron energy loss spectrum

Auger electron spectrum

Optical absorption

Photoelectron spectroscopy and optical absorption

[Jiang12]

Electronic band structure of semiconductors

[YuBook]

Mean field approaches

Remark: Kohn-Sham DFT is a many-body theory for the ground state total energy , but a mean-field approximation for single-electron excitation spectrum.

(Illustrations from G.-M. Rignanese’s talk)

The band gap problem

NiO

The origin of the DFT band gap problem

Perdew & Levy (1983); Sham & Schlueter (1983); Godby & Sham (1988)

KS HOMO-LUMO Gap ≠ Egap even with exact Exc

But for all explicit density functionals, e.g. LDA/GGA, Δxc=0

Quasi-particle theory

(courtesy of Dr. R. I. Gomez-Abal）

Concept of quasi-particles

[Mattuck]

quasi-particle theory and GW approximation

Quasi-particle equation

H. Jiang, Acta Phys.-Chim. Sin. 26, 1017(2010)

Green’s functions for single-electron Schrödinger Equations

Green’s function Green function

Definition of Green’s function (mathematically)

ˆ ( ) ( ) 0z H ψ − = r r

Consider a partial differential equation of the general form with a

certain boundary condition

z : a complex number

: a general Hermitian differential operator ˆ ( )H r

ˆ ( ) ( , '; ) ( ')z H G z δ − = − r r r r r

Green’s function G(r,r’;z) is defined as the solution of the following

equation with the same boundary condition for ψ(r):

Analytic properties of Green’s function ˆ ( ) ( , '; ) ( ')z H G z δ − = − r r r r r

*( ) ( ') ( ')n nnφ φ δ= − r r r rˆ ( ) ( ) ( )n n nH φ ε φ=r r r

* * *'( ) ( ') ( ) ( ') ( ) ( ')( , '; ) n n n n

n nn n

G z dz z z

ε εφ φ φ φ φ φ εε ε ε

= = +− − −

r r r r r rr r

G(r,r’;z) is analytic in the z-plane except at the eigenvalues of (discrete eigenvalues of ): simples poles (continuum eigenvalues of and extended states):

a branch cut

(continuum eigenvalues of and localized states): a natural boundary

( , '; ) ( , '; ), 0G G iε ε η η± ± += ± =r r r r

Green’s function for perturbation theory ˆ ( ) ( , '; ) ( ')z H G z δ − = − r r r r r

ˆ ( ) ( ) ( )

( , '; ) ( ') ', (z , )( )

( , '; ) ( ') ' ( ) (z= ) n

z H f

G z f dG f d ε

ψ

ε εψ

ε φ ε±

− = ≠

= +

r r r

r r r rr

r r r r r

0 0ˆ ( ) ( ; ) 0E H Eψ − = r r

0ˆ ( ) ( ) ( ; ) 0E H V Eψ − − = r r r 0

ˆ ( ) ( ; ) ( ) ( ; )E H E V Eψ ψ − = r r r r

0 ( , '; )G E± r r

0 0( ; ) ( ; ) ( , ') ( ') ( '; ) 'E E G V E dψ ψ ψ± ± ±= + r r r r r r r(Lippman-Schwinger equation)

Dyson’s equationIn many cases, we are interested in the perturbation expansion of

Green’s functions instead of that of wave functions

0 0

0

ˆˆ ( ) 1

ˆˆ ˆ ( ) 1

z H G z

z H V G z

− = − − =

1

0 00

1

00

1ˆ ˆ( ) ˆ

1ˆ ˆ ˆ( ) ˆ ˆ

G z z Hz H

G z z H Vz H V

−

−

= − ≡ −

= − − ≡ − −

10 0

10

ˆ ˆ( )ˆ ˆ ˆ( )

G z z H

G z z H V

−

−

= −

= − −1 1

0ˆ ˆ ˆ( ) ( )G z G z V− −= −

1 100 0

ˆ ˆ( ) (ˆ ˆ( )ˆ ˆ ˆ( ) ( )) ( )G z G z G z VGG zz G z− − = −

0 0ˆ ˆ ˆ(ˆ ˆ( ) ( )) ( )G G z G zz Vz G= + a Dyson’s equation

Time-dependent Green’s function

Time-dependent Schrödinger equation

ˆ ( ) ( , ) 0i H tt

ψ∂ − = ∂ r r

ˆ ( ) ( , ' ') ( ') ( ')i H G t t t tt

δ δ∂ − = − − ∂ r r r r r

For independent of time, , ≡ , ;Fourier transform ( ')1( , '; ') ( , '; )

2i t tG t t G e dωω ω

π+∞ − −

−∞− = r r r r

ˆ ( ) ( , '; ) ( ')H Gω ω δ − = − r r r r r

But: , ; is singular if ω is equal to any eigenvalue of !

(all in atomic units!)

Retarded and advanced Green’s function

ˆ ( ) ( , ' ') ( ') ( ')i H G t t t tt

δ δ∂ − = − − ∂ r r r r r

R/A / ( ')

( ')

1( , '; ') ( , '; )21 ( , '; )

2

i t t

i t t

G t t G e d

G i e d

ω

ω

ω ωπ

ω η ωπ

+ −+∞ − −

−∞

+∞ − −

−∞

− =

≡ ±

r r r r

r r

Retarded Green’s function

*( ) ( ')( , '; ) n n

n n

G zz

φ φε

=− r rr r

*R ( ) ( ')( , '; ) ( ) , ( ')nin n

n n

G i e t tz

ε τφ φτ θ τ τε

−= − ≡ −− r rr r

*A ( ) ( ')( , '; ) ( ) nin n

n n

G i ez

ε τφ φτ θ τε

−= −− r rr r

The use of Green’s function

R( , ) ( , '; ') ( ', ')t G t t t dΨ = − Ψr r r r r

Retarded Green’s function as the propagator

Density matrix

Eigenvalues from the poles of Green’s functions *( ) ( ')( , '; ) n n

n n

G zz

φ φε

=− r rr r

( )*

R/A

*

*( ) ( ') 1ˆ( , '; )

( ) ( ')

( ) ( ')

( , ';ˆ )

n n

n nn n

n n

n

n n n

n

G ii

i

φ φφ φω πω ε η ω ε

φ φ

δ ω ε

ρ ωπω ε

= = Ρ − ± −

−

= Ρ−

r rr r

r r

r r

r r

( )1 1ˆ i xx i x

πδη

= Ρ ±

R/A A R1 1( , '; ) Im ( , '; ) ( , '; ) ( , '; )2

G G Gρ ω ω ω ωπ π

= = − r r r r r r r r

Green’s function for many-body systems: General formalism

Outline

• Green’s functions: definition and properties

• Many-body perturbation theory based on Green’s functions

• Hedin’s equations

Representations (pictures) of quantum mechanics

Schrödinger Representation

( ) ( ) ( )

( ) ( ) ( )†

ˆ

( ) , ,

, ,

ˆS S

S

S

Si H qt

O t Oq

q t

q tqt dq

q tΨ∂ =∂

Ψ

Ψ

= Ψ

Heisenberg Representation

( )

( ) ( ) ( )( ) ( )

( ) ( )

H

†H H

H

H

H

ˆ ˆ

H

0

( )

ˆ ˆ, (assuming is independent of time)

ˆ ˆ ˆ, , ,

ˆ ,

ˆ , iHt iHtS

it

O t q q dq

e O q e H

i O q t O q t Ht

O q t

O q t

q

−

∂ =∂

= Ψ Ψ

=∂ = ∂

Ψ

Hamiltonian in terms of field operatorsField operators

Field operators in the Heisenberg representationˆ ˆ† †

ˆ ˆ

ˆ ˆ( ) ( )

ˆ ˆ( ) ( )

iHt iHt

iHt iHt

t e e

t e e

ψ ψ

ψ ψ

−

−

=

=

x x

x x

[ ] † †

†

ˆ ˆ ˆ ˆ( ), ( ') ( ), ( ') 0

ˆ ˆ( ), ( ') ( , ')

ψ ψ ψ ψ

ψ ψ δ+ +

+

= =

=

x x x x

x x x x

ˆ ( )ψ x†ˆ ( )ψ x

annihilation operator remove an electron at rcreation operator creator an electron at r

Hamiltonian of N-electron interacting systems

ˆ ˆ ˆˆ ˆ ˆ,

{ , }, : spin index

A B AB BA

s s+

≡ + ≡x r

20 ext

1ˆ ( ) ( )2

h V≡ − ∇ +x x

Definition of (one-body) Green’s function

†ˆ ˆ ˆ( , ' ') T ( ) ( ' ')G t t i N t t Nψ ψ = − x x x x

††

†

ˆ ˆ ( ) ( ' '), ' ˆ ˆ ˆT ( ) ( ' ')ˆ ˆ( ' ') ( ), '

t t t tt t

t t t tψ ψψ ψψ ψ

> = <±

x xx x

x x

N

T̂the ground state of the N-electron systemstime-ordering operator

† † †0

1ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ' ( ') ( ) ( ' ) ( ' ) ( )2

H dx t h t d d v t t t tψ ψ ψ ψ ψ ψ= + − x x x x x r r x x x x

(one-body) Green’s function

Note:

Physical significance of Green’s function (1)

't t>†ˆ ( ', ')t Nψ x1) add an electron to the system at x’ and t’

†ˆ ˆ( , ) ( ', ')N t t Nψ ψx x3) project to the ground state (measure!)

†ˆ ˆ (( ', '), ) tt Nψ ψ xx2) take an electron away from the system

at x and t

Physical significance of Green’s function (2)† †ˆ ˆ ˆ ˆ( , ' ') ( ') ( ) ( ' ') ( ' ) ( ' ') ( )iG t t t t N t t N t t N t t Nθ ψ ψ θ ψ ψ= − − −x x x x x x

't t<1) remove an electron from (add a hole to )

the system at (x,t )

ˆ ( , )t Nψ x

2) add an electron to (annihilation of the

hole) the system at (x’,t’)

†ˆ ( ', ') ˆ ( , ) Nt tψψ xx

3) project to the ground state (measure!) †ˆ ˆ( ', ') ( , )t t NN ψ ψx x

Lehmann representation (1)

Lehmann representation (2)

metallic systems: Insulating systems:

Lehmann representation (3)

Lehmann representation (4)

fs(x): Lehmann (quasi-particle) amplitudes

Spectral representation of Green’s function

Spectral function

Spectral representation of Green’s function

Alternatively,

Quasi-particles

Spectral function

Non-interacting systems

Interacting systems

(Figures from G.-M. Rignanese’s talk)

Green’s function for many-body systems: General formalism

Outline

• Green’s functions: definition and properties

• Many-body perturbation theory and Hedin’s equations

Equation of motion for GFs

Equation of motion for GFs

Similar EOS can be derived for G2 which involves G3, and so on.

Two-body Green’s function

Equation of motion for GFs: Approximations

Hartree approximation

Hartree-Fock approximation

(exchange-correlation) self-energyDefinition:

Equation of Motion

Fourier transform

Dyson’s equation

Dyson’s equation (again!)

in matrix form

Hartree approximation

Quasi-particle Equation (?)

Quasi-particles

Two major approaches to obtain approximate Σxc

Diagrammatic expansion (Wick’s theorem)

Equation of motion and functional derivatives

0 H (( ) ( ) ( , ' ) ' ( , ' ) ( ' , ' ) )) (, = ' ( )i h V G t t d dt t t G t t t tttδ δ δδ

φ ′ ′ ′′ ′ ′′ ′ ′′ ′ ′− − + − Σ − − x x x x x x x x x x xx

Hedin’s Equations

(Figures from G.-M. Rignanese’s talk)

Hedin’s Equations: Derivation (1)

adiabatic small perturbation

Hedin’s Equations: Derivation (2)

Hedin’s Equations: Derivation (3)

Hedin’s Equations: Derivation (4)

Hedin’s Equations: Derivation (5)

Hedin’s Equations: Derivation (6)

Hedin’s Equations: Derivation (7)

Hedin’s Equations: the “grand pentagon”

Ground state properties from Green’s function (1)The expectation value of one-body operator

Examples:

In general, two-body physical properties cannot be obtained

directly from one-body Green’s function.

Exception:

Galitskii-Migdal formula

Ground state properties from Green’s function (2)

GW approximation and implementations

Outline

• GW approximation

• Implementations of the GW approach

H. Jiang, Front. Chem. China 6(4): 253–268 (2011)

GW Approximation

“best G best W” : G0W0 Approximation

Hybertsen and Louie(1985); Godby, Schlüter and Sham (1986)

( ) ( ) ( ) 'xc 0 0, '; , '; ' ', ; ' '

2ii G W e dηωω ω ω ω ω

πΣ = +r r r r r r

Implementation: main ingredients

*

*

0 0 0

,

,

*

( , '; ) = ( , '; ) ( ', ; )

1 1(1 ( ') ( ')

( ')

( ))

( )

( )

( )

2

= n mm n m nn

n mn m

nm m n

m

nmn

m

f f

P

i i

i d

F

G G

ψ

ω ω ω

ω ε ε η

ω ω

ω ε ε

ω

ψψη

π

ψ

− −

′ ′ ′

Φ

− + + + − − Φ

− +

≡

x x x x x

x

xx

x

x

x x

Polarization function

Key ingredients:

How to expand the products of two orbitals the product basis

How to treat frequency dependency

( ) 0xc

* *0 0

( )'

2 '

( ) ( ) ( ') ( , '; ) ( ) ( ') '

m k k nm n

k

i j k l i

k

j k l

Wi d

W W d d

ψ ψ ω ψ ψψ ω ψ ω

π ω ω

ψψ ω ψ ψ ψ ψ ω ψ ψ

εΣ =

+ −

=

r r r r r r r r

Self-energy

sgn( )k k kiε ε η μ ε= + −

Matrix representation

Product basis:

( ), , ,ε= ≡ −cO v P W W v

c

,

Zc

*B (1( ) '

, ) ( , ')2 '

( , )n

mc m

i jnm ij nm

i j

Md

WiN

Mωω ω

π ω ω ε+∞

−∞−

Σ

=−

+ k

q k q

k q q k q

( , ; ')nmX ωk q

GW implementations

Implementation: the product basis (1)

[ ]1( ) ( ) exp ( )χ χ→ ≡ + ⋅q qGr r q G ri i

VPlanewaves

;= ( )ψ χ kk k G G

Grn nc 1/2 *

; ' ; ''

( , ) =nm n mM V C C−− −G

k G k q G GG

k q

' , '1( ) = .

| |δ

+GG G Gqq G

v ' ' '4( , ) = ( , ).

| || ' |πε ω δ ω−

+ +GG GG GGq qq G q G

P

Codes: abinit, yambo, BerkeleyGW, SaX, vasp

Atomic-like orbitals ( )1/2

1( ) = ( )αα α αχ φ⋅ + − − q R tq

Rr r R ti

c

eN

[ ] *( ) ' ( ) ( , ' ( ')) .V Vd d Xα βαβ χ χ≡ qq r rr rX r r

[ ]1 1( ) = ( ) ( ) ( )− −qX Xq S q q S q

*( ) ( ) ( ).

VS dαβ α βχ χ ≡ q qq r r r

*

,

= ( ) ( ) (( , ) )' .X α βαβα βχ χ q q

qr r q rXr

Codes: FHI-aims, FIESTA

(L)APW+lo(+LO) basis

{ } { }MBmax, '

' ' ' '( ) ( ) ( )l ll l NLl l L l l

lu r u r v rαζ αζ≤

− ≤ ≤ +⎯⎯⎯⎯→

Mixed basis

MBmax

( )

( ),

ˆ( ) ( ), MT spheres

( ) 1 , Interstitial

iNL LM

G

i ii

e v r Y

S eV

α α α

αχ

⋅ +

+ ⋅

<

∈=

∈

q R r

Rq

q G rG

G

r r

rr

Implementation: the product basis (2)

Codes: GAP, SPEX

Implementation: frequency dependence

Static approximations

Coulomb hole-screened exchange (COHSEX)

Generalized plasmon pole (GPP) model

Full frequency treatment

Imaginary frequency + analytic continuation

real frequency Hilbert transform

Contour deformation

o*

'

o*

'

SE

*

0

c

X

R ( , '; ) =

( ) ( ') ( ', ( ', ; )1( ) ( ')

1 ( ' ) ( , ';0)2

; )

( ) ( ') ( ', ;0

(

)

' , )

cc

n n nn

cc

c

n

n

n

n nn

n

WdWe

WW

ψ ψ ω ωψ ψ ωπ ω

δψ ψ

εω

ωε

∞ ′ℑ′−′−

Σ

+

≡

− ℜ

+

−

−

− ℜ

Σ

−

k k kk

k kk

k kk k

r rr r

r

r r r r

r r r r

r

r r r

r

r

r

COH ( , ')Σ r r

frequency treatment: GPP modelsPlasmon pole model for homogeneous electron gas

1Im ( , ) ( ) ( ( ))pq A q qε ω δ ω ω− ≈ −

Generalized plasmon pole models

( )1' ' '

11 ' '

'2 2 2 20'

Im ( , ) = ( ) ( )

( ) ( )2 Im ( , ) 2Re ( , ) = = .( )

A

Ad

ε ω δ ω ωωω ε ωε ω ω δ

π ω ω π ω ω

−

−∞−

−

′′+ +′ − −

GG GG GG

GG GGGG

GG

q q q

q qqq 1q

Hybertsen-Louie (HL) model

Godby-Needs (GN) model

von der Linden-Horsch (vdLH) model

Engel-Farid model

frequency treatment : Hybertsen-Louie model

( )1' ' 'Im ( , ) = ( ) ( )Aε ω δ ω ω− −GG GG GGq q q

Two parameters for each G, G’ are determined by

1) Static dielectric function ω=0

2) The f-sum rule 1 2

' '0

2'

Im ( , ) = ( )2

4 ( ) ( ')( ) ( ')| || ' |

d πω ε ω ω

π ρ

∞ − − Ω

+ ⋅ +Ω = −+ +

GG GG

GG

q q

q G q Gq G Gq G q G

1/21' ' ' '

'' 1/21

' '

( ) = ( ,0) | |2

| |( ) =( ,0)

π δ ε

ωδ ε

−

−

− Ω

Ω −

GG GG GG GG

GGGG

GG GG

q q

qq

A

frequency treatment : the GN model

( )1' ' 'Im ( , ) = ( ) ( )Aε ω δ ω ω− −GG GG GGq q q

Two parameters for each G, G’ are determined by

1) Static dielectric function ω=0

2) Inverse dielectric at a chosen imaginary frequency, ω=iωp

( )( )1/2 1 1 1 1/2' ' ' ' '

1/21 1' '1/2

' 1' '

( ) = [ ( ,0) ( ,0) ( , ) ]2

( ,0) ( , )( ) =

( ,0)

p p

pp

A i

i

π ω δ ε ε ε ω

ε ε ωω ω

δ ε

− − −

− −

−

− −

− −

GG GG GG GG GG

GG GGGG

GG GG

q q q q

q qq

q

frequency dependence: IF+AC approach

Imaginary frequency plus analytic continuation (IF-AC) o u *

2 2

2( )( , ) = 2 ( , ) ( , )

( )

BZ cc noccn n i j

ij nm nmn m n n

P iu M Mu

ε εε ε

−

−

− − × + − k k q

k k k q

q k q k q

( )( )

c20 2

2 ( , ; )( ) = .

2ε

π ε

∞ −

−

′−′Σ

′ + − k q

kq k q

k qBZm nm

nm m

iu X iuduiuu iu

p;c

;

( ) =N

p nn

p p n

aiu

iu bΣ

− kk

k

p;c

;

( ) =N

p nn

p p n

ab

ωω

Σ− k

kk

frequency dependence : the CD approach

Contour deformation (CD) approach c

F

( )( ) = .2 sgn( )

nmn

m m m

Xi diωω ω

π ω ω ε η ε ε∞

−∞

′′Σ′+ − − −

c2 20

F F

2( )( ) ( )2 ( )

( )[ ( ) ( ) ( ) ( )]

mn nm

m m

nm m m m m m

du X iuu

X

ε ωωπ ε ω

ε ω θ ε ε θ ω ε θ ε ε θ ε ω

∞ −Σ =− +

+ − − − − − −

Self-consistency: full vs approximate SCGW

Full SCGW Approx. SCGW

Faleev-van Schilfgaarde-Kotani (QSGW) scheme (PRL 2004)

Main technical parameters in GW implementation

Parameters for KS DFT:

pseudopotentials, PAW or LAPW?

basis for Kohn-Sham orbitals

Quality of product basis

Number of unoccupied states considered (P & Σc )

The integration in the Brillouin zone: the number of k/q-

points

The frequency treatment and related parameters

H. Jiang, Front. Chem. China 6(4): 253–268 (2011)

Examples for applications of the GW approach

Band gaps of semiconductors

H. Jiang: GW with HLOs-enhanced LAPW (unpublished)

M. van Schilfgaarde et al. PRL

96, 226402 (2006)

72

GW0@LDA for ZrO2 and HfO2

Jiang, H. et al. Phys. Rev. B 81, 085119 (2010)

Band gaps of MX2 and ATaO3

Jiang, H., J. Chem. Phys , 134, 204705 (2011);Jiang, H. J. Phys. Chem. C,116, 7664 (2012).

Pm-3m R3ch

H. Wang, F. Wu and H. Jiang, J. Phys. Chem. C, 115, 16180, (2011)

Ln2O3 band gaps: GW0@LDA+U vs Expt.

H. Jiang et al. Phys. Rev. Lett. 102, 126403(2009); Phys. Rev. B 86, 125115(2012).

Fully self-consistent GW for molecules

Rostgaard, Jacobsen and Thygesen, PRB 81, 085103 (2010)

GW for fullerenes

GW for CuPc

Level alignment in dye-sensitized solar cells

P. Umari et al. J. Chem. Phys. 139, 014709 (2013)C. Verdi et al, Phys. Rev. B 90, 155410 (2014)