Green’s function theory for solid state electronic band...
Transcript of Green’s function theory for solid state electronic band...
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
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)