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Theoretical approaches to the many-body electronic problem: an introduction
Lucia Reining Palaiseau Theoretical Spectroscopy Group
Theoretical approaches to the many-body electronic problem: an - - PowerPoint PPT Presentation
Theoretical approaches to the many-body electronic problem: an - - PowerPoint PPT Presentation
Theoretical approaches to the many-body electronic problem: an introduction Lucia Reining Palaiseau Theoretical Spectroscopy Group Theoretical approaches to the many-body electronic problem: an introduction Theoretical Spectroscopy: aims
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→ Theoretical Spectroscopy: aims and observations
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→ Theoretical Spectroscopy: aims and observations
550 600 650 700 704 708 712 716 720 724
Intensity
Photoelectron Kinetic Energy (eV) Photon Energy (eV)
?
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→ Theoretical Spectroscopy: aims and observations
Hψ(x1,....xN) = E ψ(x1,....xN)
?
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From Damascelli et al., RMP 75, 473 (2003)
and http://www.ieap.uni-kiel.de/surface/ag-kipp/arpes/arpes.htm
+......
→ Theoretical Spectroscopy: aims... and observations!
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From Damascelli et al., RMP 75, 473 (2003)
and http://www.ieap.uni-kiel.de/surface/ag-kipp/arpes/arpes.htm
+......
→ Theoretical Spectroscopy: aims... and observations!
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From Damascelli et al., RMP 75, 473 (2003)
and http://www.ieap.uni-kiel.de/surface/ag-kipp/arpes/arpes.htm
+......
This is not what we expect (in an i.p. picture)! → Theoretical Spectroscopy: aims... and observations!
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Valence bands Satellites
EXPO
Exp.: F. Sirotti et al., TEMPO beamline SOLEIL
→ ARPES of simple bulk silicon:
Obviously far from an i.p. picture!
Cohen and Chelikowsky: “Electronic Structure and Optical Properties of Semiconductors” Solid-State Sciences 75, Springer-Verlag 1988)
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Calculate only what you want,.....so that you can understand!
Hψn(x1,....xN) = En ψn(x1,....xN)
Want: → total energy E0 → expectation values like * density * spectral functions * dielectric function Do not want: → all many-body ψn(x1,....xN)
Vtot (ω)= ε-1(ω)Vext (ω) → Theoretical Spectroscopy: tools
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Calculate only what you want,.....so that you can understand!
Hψn(x1,....xN) = En ψn(x1,....xN)
Want: → total energy E0 → expectation values like * density * spectral functions * dielectric function Do not want: → all many-body ψn(x1,....xN)
Small systems: CI Larger: Stochastic (QMC) → Theoretical Spectroscopy: tools Vtot (ω)= ε-1(ω)Vext (ω)
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Effective quantities in an effective world A practical example, simulate zero gravity → Theoretical Spectroscopy: tools
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→ The effective quantities:
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→ The effective world:
LDA or so
Designed for density and top valence NOT for bandgaps, for example!!! Hohenberg-Kohn-Sham
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Band structure of germanium
Rohlfing et al., PRB 48, 17791 (1993)
- A. Svane, PRB 35, 5496 (1987)
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DFT: E[n] → δE /δn =0 → n0 → E0= E[n0] → F0 = F[n0] ???
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(TD)DFT point of view: moving density Change of potentials
ρ hν ρ + δρ VH+VXC VH+ δVH+VXC+δVXC
Vtot (ω)= ε-1(ω)Vext (ω)
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Excitation ? Change of potentials
RPA ρ hν ρ + δρ VH+VXC VH+ δVH+VXC+δVXC
→ Induced potentials
TDLDA, ….
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Graphene, π plasmon
- R. Hambach, Diplomarbeit and PhD thesis
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Graphene, π plasmon
- C. Kramberger et al., PRL 100, 196803 (2008)
E,k E',k' = k-q
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Graphene, π plasmon
- C. Kramberger et al., PRL 100, 196803 (2008)
U s u a l l y t h e r e i s s t r
- n
g c
- u
p l i n g i n s p e c t r
- s
c
- p
y
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GG1 G1G2
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→ Interaction leads to........... coupling
E,k E',k' = k-q Loss spectroscopy
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Exp: Eberlein et al., Phys. Rev. B 77, 233406 (2008)
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→ Interaction leads to........... coupling
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→ Interaction leads to........... coupling
Why study this? * Unexpected effects! * Guideline for experiments
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inelastic elastic inelastic Close to Bragg point
c Independent particles With induced potentials: Induced modes Ralf Hambach et al., Phys. Rev. Lett. 101, 266406 (2008)
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Experimental verification, N. Hiraoka et al., Spring8 Taiwan/Japan
Strong changes close to Bragg reflex!
Ralf Hambach et al., Phys. Rev. Lett. 101, 266406 (2008)
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→ Interaction leads to........... decay
Why study this? * Closer to experiment * Example carrier lifetime
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GG1 G1G2 Dynamic coupling difficult in TDDFT (ω)
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Effective quantities in an effective world How do we get this one? → Theoretical Spectroscopy: tools
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→ The effective quantities: → Propagators
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G(1,2) = -i <T[ψ(1)ψ†(2)]>
1=(r1,σ1,t1) Dyson equation: G =G0 + G0 Σ G
12-37
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→ The effective world:
Σ(r,r',εi)
Designed for electron addition and removal spectra (bandstructure, lifetimes, satellites,....,density,...)
Other: DMFT Σιι(ω)
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→ Σ ~ i WG “GW”
- L. Hedin (1965)
W = ε-1(ω) v
+ ….....
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GW calculations, Rohlfing et al., PRB 48, 17791 (1993)
LDA GW HF
GW today: standard for bandstructures
Bandstructure of germanium, theory versus experiment
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Inverse electron and hole lifetime in silicon
- A. Fleszar and W. Hanke, PRB 56, 10228 (1997)
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E,k E',k' = k-Q
Effect of electron and hole decay on plasmon spectra
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Effect of electron and hole decay on plasmon spectra
Cazzaniga et al;, Huotari et al.; PRB 84, 075108 and 075109 (2011)
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→ The effective quantities: → Propagators
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G(1,2) = -i <T[ψ(1)ψ†(2)]>
1=(r1,σ1,t1) Dyson equation: G =G0 + G0 Σ G
12-37
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→ The effective world:
Σ(r,r',εi)
Designed for electron addition and removal spectra (bandstructure, lifetimes, satellites,....,density,...)
Other: DMFT Σιι(ω)
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→ Σ ~ i WG “GW”
- L. Hedin (1965)
W = ε-1(ω) v
+ ….....
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- A. Tkatchenko et al., Phys. Rev. Lett. 106:118102, 2011
alanine polypeptide
Van der Waals
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P . Rinke, et al., Phys. Rev. A 70:063201, 2004.
Image states
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Molecules on surfaces
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Molecules on surfaces
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Molecules on surfaces
- C. Freysoldt, et al., Phys. Rev. Lett. 103:056803, 2009.
- J. M. Garcia-Lastra, et al, Phys. Rev. B 80:245427, 2009.
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From Damascelli et al., RMP 75, 473 (2003)
+......
Why study this? * More added value * Example multiple exciton generation
→ Interaction leads to........... additional excitations
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Valence bands Satellites
EXPO
Cohen and Chelikowsky: “Electronic Structure and Optical Properties of Semiconductors” Solid-State Sciences 75, Springer-Verlag 1988)
- M. Guzzo et al., PRL 107, 166401 (2011)
→ Interaction leads to........... additional excitations
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Valence bands Satellites
EXPO
Cohen and Chelikowsky: “Electronic Structure and Optical Properties of Semiconductors” Solid-State Sciences 75, Springer-Verlag 1988)
- M. Guzzo et al., PRL 107, 166401 (2011)
→ Interaction leads to........... additional excitations
GW not sufficient !!!
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GW:
W(r,r',ω) → one efficient plasmon Cumulant: W(r,r',ω) →series of plasmons
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Valence bands Satellites
EXPO
- M. Guzzo et al., PRL 107, 166401 (2011)
→ Interaction leads to many additional excitations
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Exp: Eberlein et al., Phys. Rev. B 77, 233406 (2008)
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McFeely et al., PRB 9, 5268 (1974)
XPS carbon 1s
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- M. Guzzo et al., Phys. Rev. B 89, 085425 (2014)
Graphite valence double plasmon: shift + broadening
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Coupling occupied and empty states: more correlation
Homogeneous Electron Gas Kas, Rehr, Reining (2014) http://arxiv.org/abs/1402.0022
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Coupling occupied and empty states: more correlation
Homogeneous Electron Gas Kas, Rehr, Reining (2014) http://arxiv.org/abs/1402.0022
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- Exp. Na, S. Huotari et al, PRL 105, 086402 (2010)
Homogeneous Electron Gas Kas, Rehr, Reining (2014) http://arxiv.org/abs/1402.0022
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→ Understanding? Our theory is: decomposition into different experiments!
PES IXS
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“Plasmon” contributions also interesting and accessible in TMOs: VO2 CuO GW gap 1.7 – 4.2 eV
- Exp. 1.3 +/- 0.3 eV
Gatti, Panaccione, Reining (PRL 2015) Roedl, Sottile, Reining (PRB 2015)
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→ Gap theo. < ~ 2 eV; exp. 1.4 +/- 0.3 eV
- C. Roedl et al., 2016
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→ Theory and Experiment
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→ Theory and Experiment
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→ Theory and Experiment
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→ Theory and Experiment
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e-h problem: Bethe-Salpeter equation Dressed hole Dressed electron e-h interaction → Electron-hole correlation
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Larson et al., PRL 99, 026401 (2007) Exciton: Lee, Hsueh, Ku, PRB 82, 081106 (2010)
- V. Olevano et al. (2000)
(bulk silicon 1998)
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Larson et al., PRL 99, 026401 (2007) Exciton: Lee, Hsueh, Ku, PRB 82, 081106 (2010)
- V. Olevano et al. (2000)
(bulk silicon 1998)
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Hydrogen series in absorption spectrum of solid argon
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- F. Sottile et al., Phys Rev. Lett 91, 056402 (2003).
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Exciton dispersion in LiF
- M. Gatti and F. Sottile, Phys. Rev. B 88, 155113
- Exp. P. Abbamonte et al., Proc. Natl. Acad. Sci. USA 105, 12159 (2008).
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Signifjcant excitonic efgect Very good agreement with experiment
PhD thesis Igor Reshetnyak (23.9.2015)
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Rohlfing and Louie, PRL 81, 2312 (1998)
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Strongly bound exciton visible
Mixed Dynamic Structure Factor
q=(0,0.25,0.25), G=(0,0,0), G’=(1,1,1)
87
PhD thesis Igor Reshetnyak (23.9.2015)
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What can we do with it? For example, induced charges
In linear response:
88
Ralf Hambach Giulia Pegolotti Claudia Roedl Igor Reshetnyak (exchange)
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Plane-wave external potential Excitonic efgects visible
Induced Charges
[1] P . Abbamonte et al. Phys. Rev. Lett. 2004. [2] P . Abbamonte et al. Advanced Materials 2010.
PhD thesis I. Reshetnyak
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RPA BSE
Consequences of excitons?
At 14.1 eV PhD thesis I. Reshetnyak
The whole matrix follow excitations in real space and time →
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Excitonic efgects in photoemission satellites
91
[1] Marisa Scrocco Phys. Rev. B, 1985.
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Overall comparison to experiments
92
[1] S. P . Kowalczyk et al. Phys. Rev. B, 1974
Li1s F2s F2p
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Analysis
[1] Marisa Scrocco Phys. Rev. B, 1985.
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Analysis
[1] Marisa Scrocco Phys. Rev. B, 1985.
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H2
+
Is life that simple?
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H2
+
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H2
+
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H2
+
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H2
+
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H2
+
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H2
+
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H2
+
Romaniello, Guyot, Reining, J Chem Phys 131, 154111 (2009)
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Correlation beyond mean field response
H2
+
gap
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Is life that simple?
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The GW image
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The GW image
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See also Romaniello, Guyot, Reining, J Chem Phys 131, 154111 (2009) And Romaniello, Bechstedt, Reining, PRB 85, 155131 (2012)
The T-matrix image
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Springer, Aryasetiawan, Karlsson, PRL 80, 2389 (1998)
More than academic: 6 eV satellite in Ni = hole-hole
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Effective quantities in an effective world A practical example, simulate zero gravity → Theoretical Spectroscopy: tools
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Suggested Reading
Onida, G., Reining, L., and Rubio, A., “Electronic excitations: density-functional versus many-body Greens-function approaches,” Rev. Mod. Phys. 74, 601, 2002. Review of ab initio calculations
- f electronic excitations with accent on optical properties and a comparison between Bethe–
Salpeter and TDDFT Strinati, G., “Application of the Green’s function method to the study of the optical-properties of semiconductors,” Rivista del Nuovo Cimento 11, 1, 1988. Pedagogical review of the theoretical framework underlying today’s Bethe–Salpeter calculations. Derivation of the main equations and link to spectroscopy. R.M. Martin, L. Reining, D.M. Ceperley, “Interacting Electrons: Theory and Computational Approaches, Cambridge May 2016 New book containing many-body perturbation theory, DMFT and QMC
- L. Hedin, “On correlation effects in electron spectroscopies and the GW approximation,”
- J. Phys. C 11:R489–528, 1999. Short review, very good for photoemission!
- F. Aryasetiawan and O. Gunnarsson, “The GW method,” Rep. Prog. Phys. 61:237–312,
1998; and:
- W. G. Aulbur, L. Jonsson, and J. W. Wilkins, “Quasiparticle calculations in solids,” Solid
State Phys. 54:1–218, 2000; Two nice and quite complete reviews on GW
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Palaiseau Theoretical Spectroscopy Group & friends
Matteo Guzzo, Ralf Hambach, Igor Reshetnyak, Claudia Roedl, Lorenzo Sponza, Sky (Jianqiang) Zhou, Francesco Sottile, Matteo Gatti, Christine Giorgetti, Hansi Weissker, Lucia Reining Toulouse: Pina Romaniello, Arjan Berger
- U. Washington: John Rehr, Joshua Kas
Synchrotron SOLEIL: Fausto Sirotti, Matthieu Silly Synchrotron ESRF: Simo Huotari, Giulio Monaco Synchrotron ELETTRA: Giancarlo Panaccione