manybody perturbation theory: from fundamental ideas to...
TRANSCRIPT
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Manybody perturbation theory: From fundamental ideas to use in practice
Lucia ReiningPalaiseau Theoretical Spectroscopy Group
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1) Aims2) Reminder TDDFT: induced potentials3) Add a particle: GW
* exchange* screening* dynamical screening
4) Neutral excitations in the (quasi)particle world: BSE * changes in transition energies
* mixing of transitions5) Problems6) Summary
Manybody perturbation theory:
From fundamental ideas to use in practice
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Ab initio calculations of electronic excitations
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550 600 650 700704
708
712
716
720
724Intensity
Photoelectron Kinetic Energy (eV)
Ph
oto
n E
ner
gy
(eV
) ?
Ab initio calculations of electronic excitations
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H(x1,....x
N) = E (x
1,....x
N)
?
Ab initio calculations of electronic excitations
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realisticfeasible
easy to interpret
Calculations
Ab initio calculations of electronic excitations
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hν
Independent electrons and transitions
Im [0] ~ vc |<v|D|c>|2 (EcEv)
v
c
Energies and wavefunctions often from DFT – Kohn Sham
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PCM: Reversible phase transition
rapid transition: ~10100ns large optical & electric contrast
although already applied still not
well understood
GeSbTe alloys:200nm
W. Welnic et al., Phys. Rev. Lett. 98, 236403 (2007)
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calculations & experiment
spectroscopy on polycrystalline thin films of GeTe optical spectra calculations (RPA)
change in local order results in change of optical absorption.
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Origin of strong change in absorption
No significant changes in jdos
Absorption given by Fermi's golden rule:
Matrix elements define
optical contrast!
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Silicon:
3 – 5 eV
20 eV
Induced potentials
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Excitation ?
Change of potentials
h
V
H+V
XC VH+ V
H+V
XC+V
XC
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Excitation ?
Change of potentials
Induced Hartree: longrange and local field effects
RPA
h
V
H+V
XC VH+ V
H+V
XC+V
XC
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TDDFTRPA: = 0 + 0 [ v ]
v (q+G) ~ 1/|q+G|2
G=0 → plasmonG ǂ 0 → crystal local field effects
VH/
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Graphene, plasmon
R. Hambach, Diplomarbeit
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Graphene, plasmon
C. Kramberger et al., PRL 100, 196803 (2008)
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TDDFT = 0 + 0 [ v + fxc ]
Vxc/
TDLDA: VxcLDA(r,t)/ (r',t') = (rr')(tt')dVxc/d
VH/
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RPA and TDLDA: sometimes quite wrong!
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Move to another framework
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RPA and TDLDA: sometimes quite wrong!
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EcEv (minimum, eV)
DFTKS0.554.265.38.2
SiliconDiamondMgOAr
Exp.1.175.487.8314.2
KS underestimates gaps (transition energies)
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v
c
Hole (N1) electrons
“Real” (quasi…) electrons:
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Exchange
Relaxation dynamical correlations
(∇ 2/2 + Vext + VH + Vxc(r)) |n> = En |n>
(∇ 2/2 + Vext + VH + ) |n> = En |n>
Vxc(r) → (r,r’,En)
= iGW L. Hedin, PR 1965W= 1v !!!
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G(1,2) = i <T[†(2)]> 1=(r1,
1,t
1)
Dyson equation: G=G0 + G
0 G
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From Damascelli et al., RMP 75, 473 (2003)
A()~Im[G()]
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Interaction = variation of “potential”
What does the system do (create eh pairs....)
VH(12)= i(12)G(33+)v
c(31)
x(12)= iG(12+)v
c(21)
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Interaction = variation of “potential”
What does the system do (create eh pairs....)
VH(12)= i(12)G(33+)v
c(31)
x(12)= iG(12+)v
c(21)
System does nothing: HFOnly classical part of : L→i,
xc→ iG(v
c + v
c v
c) =iGW
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What is so good about GW?
→ Fock exchange localizationL. Hedin, PR 139, A796 (1965)
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Away from the LDA starting point!→ S. V. Faleev, M. van Schilfgaarde, and T. Kotani, PRL 93, 126406 (2004).→ Bruneval, Vast, and Reining, Phys. Rev. B 74, 045102 (2006).COHSEX→ T. Miyake et al., Phys. Rev. B 74, 245231 (2006).LDA+U→ F. Fuchs, et al., Phys. Rev. B 76, 115109 (2007). LDA+U, hybrids→ Hong Jiang et al., , Phys. Rev. Lett. 102, 126403 (2009). LDA+U→ P. Rinke et al., New J. Phys. 7, 126 (2005). KSEXX→ ...........................
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Improve QP energies and wavefunctions
van Schilfgaarde, Kotani, Faleev,Phys. Rev. Lett. 96, 226402 (2006)
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rutile monoclinic
Even qualitatively important VO2
Matteo Gatti et al.
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T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
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T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
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T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
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M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
LDA insulator → metal ....
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M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
G0W
0 insulator → metal !!!!!
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M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
sc GW insulator = insulator !!!
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What is so good about GW?
→ Fock exchange
→ Screening
v
c
localization
relaxation
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...of practical importance: example photovoltaics
J. Vidal et al., collaboration EDF, PRL 104, 056401 (2010)
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Bandgaps in function of structure (CuS)
Hybrids ~ approximate GW with almost fixed screening
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What is so good about GW?
→ Fock exchange
→ Screening
→ Screening (dynamical)
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W() leads to imaginary part
Broadening (lifetime), satellites
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Screening (dynamical): W
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MetalInsulator
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R. Hambach, PhD thesis
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H. Abe et al., Jpn. J. Appl. Phys. 36, 165 (1997)
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M. Gatti et al., 2010
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Screening (dynamical): W
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EcEv (minimum, eV)
DFTKS0.554.265.38.2
SiliconDiamondMgOAr
Σ =iGW1.195.647.814.0
Exp.1.175.487.8314.2
GW contains screening of hole (or electron) correct energies et al.
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Even good bandstructure → sometimes quite wrong!
But back to absorption.......
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Absorption ?
v
c
Electronhole interaction
Excitonic effects
BetheSalpeter Equation
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Solve a 2particle equation: Bethe Salpeter Eq.
Dysonlike equation: BSE
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TDDFT = 0 + 0 [ v + fxc ]
Vxc/
TDLDA: VxcLDA(r,t)/ (r',t') = (rr')(tt')dVxc/d
VH/
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Solve a 2particle equation: Bethe Salpeter Eq.
Dysonlike equation: BSE
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Im [] ~ vc |<v|D|c>|2 (EcEv)
(Hel + Hhole + Helhole ) A = E A
Im [] ~ | vc<v|D|c> Avc|2 (E)
>Mixing of transitions
>Modification of excitation energies
GW + Bethe Salpeter Equation
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Im [] ~ vc |<v|D|c>|2 (EcEv)
(Hel + Hhole + Helhole ) A = E A
Im [] ~ | vc<v|D|c> Avc|2 (E)
>Mixing of transitions
>Modification of excitation energies
From VH/ + /G
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F. Bruneval et al., PRL 97, 267601 (2006))
Excitons in Cu2O
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V. Garbuio, et al., PRL. 97, 137402 (2006)
Excitons in water
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Excitons in Na4
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Electronhole interaction for good absorption spectra
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Is there anything BAD about GW?
→ Fock exchange
→ Screening
→ Screening (dynamical)
→ Widely valid and efficient
→ Room for speedup
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G(1,2) = i <T[†(2)]> 1=(r1,
1,t
1)
Dyson equation: G=G0 + G
0 G
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More consistent in orders of W
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Interaction = variation of “potential”
What does the system do (create eh pairs....)
System does nothing: HFOnly classical part of : L→i,
xc→ iGW
Additional particle interacts in a classical way
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2 site 1 electron Hubbard modelP. Romaniello et al., J. Chem. Phys 2009, 131, 154111
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2 site 1 electron Hubbard model
We should consider 2 particles at a time
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Interaction = variation of “potential”
What does the system do (create eh pairs....)
System does nothing: HFOnly classical part of : L→i,
xc→ iGW
Better : e.g. Tmatrix approach
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Springer, Aryasetiawan, Karlsson, PRL 80, 2389 (1998)
6 eV satellite in Ni: 2hole bound state
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We are used to solve 2particle equations: BSE
Dysonlike equation: BSE
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5) Summary
Keywords:
* quasiparticles * induced potentials* screening* localization* satellites* Dyson equations* .....breakdowns....
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http://www.etsf.eu
Palaiseau Theoretical Spectroscopy Group
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4. Look at the total energy
Adiabatic connection fluctuation dissipation theorem (ACFDT)
........in DFT framework
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Total energy in GW framework
.......evaluated at G=Gs, equivalent to RPA ACFDT
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RPA+ : because RPA has deficiencies
Uniform gas correlation energy too negative.Can be corrected by LDA like term. S. Kurth and J. P. Perdew, Phys. Rev. B 59, 10 461(1999); M. Lein, E. K. U. Gross, and J. P. Perdew,
Phys. Rev. B 61, 13 431 (2000).
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Sum over empty states: how to speed up
Replace i by one representative energy?
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Bruneval, Gonze, PRB 78, 085125 (2008). SiC
From sum rules
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Argon, no empty states summed. Berger et al., PRB 82, 041103R 2010
Very simple to be implemented, imediate speedup.
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C. Kramberger et al., PRL 100, 196803 (2008)
Isolated carbon nanotube, plasmon
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Nanotubes and graphene, plasmon dispersion
C. Kramberger et al., PRL 100, 196803 (2008)