The GW approximation in less than 60 minutes F. Bruneval Service de Recherches de Métallurgie Physique CEA, DEN
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Outline
I. Standard DFT suffers from the band gap problem II. Introduction of the Green's function III. The GW approximation IV. The GW code in ABINIT and the G0W0 method V. Some applications F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Standard DFT has unfortunately some shortcomings
band gap
Band gap problem! after van Schilfgaarde et al PRL 96 226402 (2008)
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
A pervasive problem Effective masses for transport in semiconductors
Optical absorption
onset
Defect formation energy, dopant solubility
Photoemission
Exp.
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
How do go beyond within the DFT framework? Not easy to find improvement within DFT framework There is no such thing as a perturbative expansion Perdew's Jacob's ladder does not help for the band gap
after J. Perdew JCP (2005).
Need to change the overall framework! F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Outline
I. Standard DFT suffers from the band gap problem II. Introduction of the Green's function III. The GW approximation IV. The GW code in ABINIT and the G0W0 method V. Some applications F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Many-body perturbation theory
Historically older than the DFT (1940-50's)! Big names: Feynman, Schwinger, Hubbard, Hedin, Lundqvist
Green's functions = propagator
G r t , r ' t '=
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
The Green's function
∣N ,0 〉
Exact ground state wavefunction:
Creation, annihilation operator:
1
2
†
r t ∣N ,0 〉
†
r t , r t
is a (N+1) electron wavefunction not necessarily in the ground state
†
r ' t ' ∣N ,0 〉 is another (N+1) electron wavefunction Let's compare the two of them! F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Green's function definition
†
〈 N ,0∣ r t r ' t ' ∣N , 0 〉 2
1
e
=i G r t , r ' t '
for
tt '
Mesures how an extra electron propagates from (r't') to (rt).
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Green's function definition
†
〈 N ,0∣ r ' t ' r t ∣N , 0 〉 1
2
h
for
=i G r ' t ' , r t
t ' t
Mesures how a missing electron (= a hole) propagates from (rt) to (r't').
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Final expression for the Green's function
i G r t , r ' t ' = † 〈 N ,0∣T [ r t r ' t ' ]∣ N , 0 〉 time-ordering operator
e
G r t , r ' t '=G r t , r ' t ' h −G r ' t ' , r t Compact expression that describes both the propagation of an extra electron and an extra hole F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Lehman representation
i G r , r ' , t−t ' = 〈 N ,0∣T [ r t r ' t ' ]∣N ,0 〉 †
Closure relation
∑N ,i ∣ N , i 〉 〈 N ,i∣ Lehman representation:
G r , r ' , =∑ i
where
{
f i r f r ' −i±i
E N 1,i−E N ,0 i= E N ,0−E N −1,i
F. Bruneval
∗ i
Exact excitation energies!
1st Yarmouk school, Irbid 4 november 2010
Related to photoemission spectroscopy Ekin
hν
Energy conservation:
before
after
h E N ,0=E kinE N −1, i Quasiparticle energy: F. Bruneval
i=E N ,0−E N −1, i=E kin−h 1st Yarmouk school, Irbid 4 november 2010
And inverse photoemission spectroscopy hν
Ekin
Energy conservation:
before
after
E kinE N ,0=h E N 1,i Quasiparticle energy: F. Bruneval
i=E N 1,i−E N ,0=E kin −h 1st Yarmouk school, Irbid 4 november 2010
Other properties of the Green's function Galitskii-Migdal formula for the total energy:
E total =
1 d Tr [ −h0 Im G ] ∫ −∞
Expectation value of any 1 particle operator (local or non-local)
〈 O 〉=lim Tr [ OG ] t t '
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Outline
I. Standard DFT suffers from the band gap problem II. Introduction of the Green's function III. The GW approximation IV. The GW code in ABINIT and the G0W0 method V. Some applications F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
How to calculate the Green's function?
Feynman diagrams
Hedin's functional approach
F. Bruneval
PRA (1965).
1st Yarmouk school, Irbid 4 november 2010
Hedin's coupled equations 6 coupled equations:
1=r 1 t 1 1
2= r 2 t 2 2
G1,2=G 0 1,2∫ d34 G0 1,3 3,4G 4,2 1,2=i ∫ d34 G 1,3W 1,4 4,2 ,3 1,2 ,3= 1,2 1,3∫ d 4567
Dyson equation self-energy
1,2 G 4,6G5,7 6,7 ,3 G 4,5
vertex
0 1,2=−i ∫ d34 G 1,3G 4,1 3,4 ,2
polarizability
1,2=1,2−∫ d3 v 1,3 0 3,2
dielectric matrix
−1
W 1,2=∫ d3 1,3 v3,2
F. Bruneval
screened Coulomb interaction
1st Yarmouk school, Irbid 4 november 2010
Simplest approximation
1,2=i G1 ,2v 1,2
t
Fock exchange
Dyson equation:
v 1,2
G=G0G 0 G G=G0G 0 G0...
G1 ,2
Not enough: Hartree-Fock is know to be quite bad for solids F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Hartree-Fock approximation for band gaps
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Hedin's coupled equations 6 coupled equations: G1,2=G 0 1,2∫ d34 G0 1,3 3,4G 4,2 1,2=i ∫ d34 G 1,3W 1,4 4,2 ,3 1,2 ,3= 1,2 1,3∫ d 4567
Dyson equation self-energy
1,2 G 4,6G5,7 6,7 ,3 G 4,5
0 1,2=−i ∫ d34 G 1,3G 4,1 3,4 ,2 1,2=1,2−∫ d3 v 1,3 0 3,2 −1
W 1,2=∫ d3 1,3 v3,2
F. Bruneval
screened Coulomb interaction
1st Yarmouk school, Irbid 4 november 2010
Hedin's coupled equations 6 coupled equations: G1,2=G 0 1,2∫ d34 G0 1,3 3,4G 4,2 1,2=i ∫ d34 G 1,3W 1,4 4,2 ,3 1,2 ,3= 1,2 1,3∫ d 4567
Dyson equation self-energy
1,2 G 4,6G5,7 6,7 ,3 G 4,5
0 1,2=−i ∫ d34 G 1,3G 4,1 3,4 ,2 1,2=1,2−∫ d3 v 1,3 0 3,2 −1
W 1,2=∫ d3 1,3 v3,2
F. Bruneval
screened Coulomb interaction
1st Yarmouk school, Irbid 4 november 2010
Hedin's coupled equations 6 coupled equations: G1,2=G 0 1,2∫ d34 G0 1,3 3,4G 4,2 1,2=i ∫ d34 G 1,3W 2 1,4 2 4,2 ,3 1,2 ,3= 1,2 1,3∫ d 4567
Dyson equation self-energy
1,2 G 4,6G5,7 6,7 ,3 G 4,5
0 1,2=−i ∫ d34 G 1,3G 3,4 ,2 2 4,1 2 1,2=1,2−∫ d3 v 1,3 0 3,2 −1
W 1,2=∫ d3 1,3 v3,2
F. Bruneval
screened Coulomb interaction
1st Yarmouk school, Irbid 4 november 2010
Here comes the GW approximation
1,2=i G1,2W 1,2
GW approximation
0 1,2=−i G1,2G 2,1
RPA approximation
1,2=1,2−∫ d3 v 1,3 0 3,2
W 1,2=∫ d3 −1 1,3 v3,2
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
What is W? Interaction between electrons in vacuum: 2
1 e V r , r ' = 4 0 ∣r −r '∣ Interaction between electrons in a homogeneous polarizable medium: 2
1 e W r , r '= 4 0 r ∣r−r '∣ Dielectric constant of the medium
Dynamically screened interaction between electrons in a general medium: 2
−1
e r , r ' ' , W r , r ' , = d r ' ' ∫ 4 0 ∣r ' '−r '∣ F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
W is frequency dependent W can measured directly by Inelastic X-ray Scattering W q=0.80 a.u , Silicon Plasmon frequency
ω [eV] Zero below the band gap F. Bruneval
H-C Weissker et al. PRB (2010) 1st Yarmouk school, Irbid 4 november 2010
GW has a “super” Hartree-Fock GW Approximation
Hartree-Fock Approximation x r 1, r 2 =
xc r 1, r 2, =
i d ' G r 1, r 2, ' v r 1, r 2 ∫ 2 −∞
i d ' G r 1, r 2, ' W r 1, r 2, ' ∫ 2
= bare exchange
x r 1, r 2 Bare exchange
c r 1, r 2, + correlation GW is nothing else but a “screened” version of Hartree-Fock.
F. Bruneval
Non Hermitian dynamic
1st Yarmouk school, Irbid 4 november 2010
Summary: DFT vs GW
Electronic density
r Local and static
exchange-correlation potential
v xc r Approximations:
LDA, GGA, hybrids
Green's function
G r t , r ' t ' Non-local, dynamic Depends onto empty states
exchange-correlation operator = self-energy
xc r , r ' , GW approximation
GW r t , r ' t '=iG r t , r ' , t 'W r t , r ' t ' F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
GW approximation gets good band gap
No more a band gap problem ! after van Schilfgaarde et al PRL 96 226402 (2008)
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Outline
I. Standard DFT suffers from the band gap problem II. Introduction of the Green's function III. The GW approximation IV. The GW code in ABINIT and the G0W0 method V. Some applications F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Available GW codes
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Available GW codes
has a GW code inside F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Code history Rex Godby
PRL 1986
Lucia Reining Giovanni Onida Valerio Olevano
Marc Torrent G.-M. Rignanese
Fabien Bruneval
M. Giantomassi
~ 1993
2001
Today F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
How to get G? Remember the Lehman representation:
f i r f r ' −i±i
G r , r ' , =∑ i
where the
∗ i
f i r and the i are complicated quantities
But for independent electrons like Kohn-Sham electrons:
G r , r ' ,=∑ KS
KS i
KS ∗ i KS i
r
r'
− ±i
i
This can be considered as the best guess for G One can get F. Bruneval
W
and
GW
1st Yarmouk school, Irbid 4 november 2010
GW as a perturbation with respect to LDA GW quasiparticle equation:
[ h0 xc ]∣ 〉= ∣ 〉 GW i
GW i
GW i
GW i
KS equation:
[h
0
v
Approximation :
F. Bruneval
LDA xc
]∣ 〉= ∣ 〉
GW i
LDA i
≈
LDA i
LDA i
LDA i
1st Yarmouk school, Irbid 4 november 2010
GW as a perturbation with respect to LDA GW quasiparticle equation:
〈 ∣[ h LDA i
0
xc
GW i
]∣
LDA i
〉=
GW i
KS equation:
〈 ∣[ h v ]∣ 〉= LDA i
GW i
−
F. Bruneval
LDA i
=〈
LDA xc
0
LDA i
∣[ xc
LDA i
GW i
−v
LDA i
LDA xc
]∣ 〉 LDA i
1st Yarmouk school, Irbid 4 november 2010
Linearization of the energy dependance
GW i
−
LDA i
=〈
∣[ xc
LDA i
GW i
−v
LDA xc
]∣ 〉 LDA i
Not yet known
Taylor expansion: GW i
xc
= xc
LDA i
GW i
LDA i
−
∂ xc ... ∂
Final result:
GW i
=
LDA i
where F. Bruneval
Z i 〈
∣[ xc
LDA i
LDA i
∂ xc Z i =1/ 1− ∂
−v
LDA xc
]∣ 〉 LDA i
1st Yarmouk school, Irbid 4 november 2010
Quasiparticle equation A typical ABINIT ouptput for Silicon at Gamma point k = Band 4 5
0.000 0.000 0.000 E0 SigX SigC(E0) 0.506 -11.291 -12.492 0.744 3.080 -10.095 -5.870 -3.859
E^0_gap E^GW_gap
GW i
=
Z dSigC/dE Sig(E) 0.775 -0.291 -11.645 0.775 -0.290 -9.812
E-E0 -0.354 0.283
E 0.152 3.363
2.574 3.212
LDA i
F. Bruneval
Z i 〈
∣[ xc
LDA i
LDA i
−v
LDA xc
]∣ 〉 LDA i
1st Yarmouk school, Irbid 4 november 2010
Flow chart of a typical GW calculation DFT
LDA i
,
LDA i
occupied AND empty states
calculate W
If self-consistent GW GW , i i
F. Bruneval
Eigenvalues
calculate G * W
GW i
k
1st Yarmouk school, Irbid 4 november 2010
Outline
I. Standard DFT suffers from the band gap problem II. Introduction of the Green's function III. The GW approximation IV. The GW code in ABINIT and the G0W0 method V. Some applications F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
GW approximation gets good band gap
No more a band gap problem ! after van Schilfgaarde et al PRL 96 226402 (2008)
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Clusters de sodium
−
Na 4 e ⇔ Na 4 4
E 0 Na 4 −E 0 Na =
{
HOMO , Na4 LUMO , Na4
+
Na4 /Na4 Bruneval PRL (2009) F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Calculs de défauts avec l'approximation GW Calcul d'un système à 215 atomes Carbure de Silicium cubique 3C-SiC
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Barrière de transformation
Exp : EA = 2.2 ± 0.3 eV
Bruneval et Roma soumis (2010) F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
Band Offset at the interface between two semiconductors Very important for electronics!
Example: Si/SiO2 interface for transistors
GW correction with respect to LDA F. Bruneval
R. Shaltaf PRL (2008). 1st Yarmouk school, Irbid 4 november 2010
Summary ●
●
●
The GW approximation solves the band gap problem! The calculations are extremely heavy, so that we resort to many additional technical approximations: method named G0W0 The complexity comes from ●
Dependance upon empty states
●
Non-local operators
●
●
Dynamic operators that requires freq. convolutions
There are still some other approximations like the Plasmon-Pole model... that I'll discuss during the practical session...
F. Bruneval
1st Yarmouk school, Irbid 4 november 2010
