Ab Initio theories to predict ARPES Hedin's GW and beyond

Transcription

1 Ab Initio theories to predict ARPES Hedin's GW and beyond Valerio Olevano Institut NEEL, CNRS, Grenoble, France and European Theoretical Spectroscopy Facility

2 Many thanks to: Matteo Gatti, Pierre Darancet, Fabien Bruneval, Francesco Sottile and Lucia Reining Institut NEEL, CNRS, Grenoble, France and LSI, CNRS - CEA Ecole Polytechnique, Palaiseau France

3 Résumé Motivation: Electronic Excitations and Spectroscopy Many-Body Perturbation Theory and the Hedin's GW approximation -> ARPES Non-Equilibrium Green's Function (NEGF) theory, GW approximation -> e-e in Quantum Transport MBPT using the Density-functional concept: vertex corrections beyond GW. Generalized Sham-Schlüter Equation and frequencydependent effective local potentials. Conclusions

4 The Ground State Ab initio DFT theory well describes (error 1~2%): Ground State Total Energy and Electronic Density Atomic Structure, Lattice Parameters Elastic Constants Phonon Frequencies that is, all the Ground State Properties. Vanadium Oxide, VO2 lattice parameters a b c α DFT-LDA nlcc DFT-LDA semic Å Å Å Å Å Å EXP [Longo et al.] ± Å ± Å ± Å ± M. Gatti et al. To be published

5 Citation Statistics from 110 years of Physical Review S. Redner, Physics Today June 2005, 49. DFT Standard Model of Condensed Matter DFT foundation Jellium xc calc. and param. for LDA LAPW LMTO k-points for BZ

6 Excited States But can DFT describe the Excited States, such as: Band Gap, Band Plot Metal/Insulator character Spectral Function? Vanadium Oxide, VO2 From H. Abe et al, Jpn. J. Appl. Phys (1997)

7 Excited States Answer: NO! DFT cannot in principle describe excited states, band gap and so on! (and it cannot be blamed if it does not succeed) Nevertheless, be careful: DFT for electronic structure -> photoemission spectroscopy DFT for optical spectroscopy DFT of superconductivity -> superconductivity gap DFT-NEGF -> quantum transport

8 Why we need ab initio theories to calculate spectra 1)To understand and explain observed phenomena 2)To offer experimentalists reference spectra 3)To predict properties before the synthesis, the experiment

9 Excited States Ab initio Theories HF (Hartree-Fock), CI (Configuration Interaction) QMC (Quantum Montecarlo) TDDFT (Time-Dependent Density-Functional Theory) MBPT (Many-Body Theory) in the Approximation: GW Photoemission NEGF (Non-Equilibrium Green's Functions Theory) Quantum Transport

10 MBPT and the GW approximation vs ARPES Photoemission Spectroscopy (Band Gap, Band Plot, Spectral Function)

11 Photoemission hν ehν c e- c v de tec tor direct photoemission 1 band gap A= ℑ G band plot hν - e sample v inverse photoemission LURE, Orsay hν e- sample Valerio Olevano, LEPES CNRS, Grenoble

12 Calculating the Band-Gap: inadequacy of HF or DFT HF always overestimates the bandgap. The Kohn-Sham energies have not an interpretation as removal/addition energies (Kopman Theorem does not hold). If we use them, however we see they are better than HF but the band gap is always underestimated. Need to go beyond: MBPT and GW! Silicon Germanium Diamond MgO Sn HF 5,6 4,2 12,10 2,60 A. Svane, PRB 1987 DFT-LDA 0,55 0 4,26 5,3 0 EXP 1,17 0,7 5,48 7,83 0 Valerio Olevano, LEPES CNRS, Grenoble

13 What is the MBPT? Many-Body Perturbation Theory is a Quantum Field Theory, based on second quantization of operators and a Green s function formalism. Advantages of the Field-Theoretic treatment: Avoids indices running on the many particles; Fermionic antisymmetrization automatically imposed; Treats systems with varying number of particles; Opens to Green s functions or Propagators which have condensed inside all the Physics (all the observables) of the system. Spectral Function A(k,ω) = Im G(k,ω) G(x1,x2) instead of Ψ(x1,...,xN) Valerio Olevano, LEPES CNRS, Grenoble

14 MBPT in brief Many-Body Perturbation Theory does not work as a Perturbation Theory -the perturbation is not small 1st order MBPT = Hartree-Fock; 2nd order not small, the series does not converge -> need to resort to complicated partial resummations of diagrams; Better functional and iterative methods: Hedin equations. Iterative solution of Hedin equations = exact solution of the problem! Valerio Olevano, LEPES CNRS, Grenoble

15 Hedin Equations (PR 139, 3453 (1965)) G=G 0 G 0 G W=v v W M =i G W = i G G M =1 GG G G Σ Γ W Π Valerio Olevano, LEPES CNRS, Grenoble So far, nobody has solved Hedin Equations for a real system Need for approximations

16 Hedin Equations: GW approximation G=G 0 G 0 G W=v v W M =i G W = i G G M =1 GG G G Σ Γ=1 W Π Valerio Olevano, LEPES CNRS, Grenoble Reviews on GW: F. Aryasetiawan and O. Gunnarsson, RPP 1998 W.G. Aulbur, L. Jonsson and J.F. Wilkins, 1999

17 Hedin's GW Approximation for the Self-Energy Dynamical Screened Interaction W GW Self-Energy GW x 1, x 2 =i G x 1, x 2 W x 1, x 2 1 Green Function or Electron Propagator G Hartree-Fock Self-Energy x x 1, x 2 =i G x 1, x 2 v x 1, x 2 Valerio Olevano, LEPES CNRS, Grenoble Bare Coulombian Potential v 2

18 Quasiparticle Energies KS energies (no physical meaning) [ ] 1 2 r v ext r v H r i r v xc r i r = Ki S i r 2 Kohn-Sham equation Exchange-Correlation potential (local) [ ] 1 2 F r v ext r v H r i r dr ' x r, r ' i r ' = H i r i 2 Hartree-Fock equation Exchange (Fock) operator (non-local) QP energies [ ] 1 2 QP QP r v ext r v H r i r dr ' r, r ', = i i r ' = i i r Quasiparticle 2 equation Self-Energy (non-local and energy dependent) Valerio Olevano, LEPES CNRS, Grenoble

19 GW and the Photoemission Band Gap Silicon Germanium Diamond MgO α-sn HF 5,6 4,2 12,10 2,60 DFT-LDA 0,55 0 4,26 5,3 0 GW 1,19 0,6 5,64 7,8 EXP 1,17 0,7 5,48 7,83 0 Our calculation but reproducing: M.S. Hybertsen and S.G. Louie (1986) R.W. Godby, M. Schlueter and L. J. Sham (1987) The GW Approximation corrects the LDA band-gap problem (underestimation) and the HF overestimation and it is in good agreement with the Experiment. The GW Approximation correctly predicts electron Addition/Removal excitations (Photoemission Spectroscopy). Valerio Olevano, LEPES CNRS, Grenoble

20 GW and the Photoemission Band Gap Adapted from Schiffelgard et al. PRL 2006

21 GW band plot Graphite J. Serrano et al, unpublished Valerio Olevano, LEPES CNRS, Grenoble

22 GW spectral function GW Γ 1v -> Γ'25v Γ'25v -> Γ 15c 3.23 Γ'25v -> Γ'2c 3.96 V. Olevano, unpublished EXP 12.5 ± Valerio Olevano, LEPES CNRS, Grenoble

23 Vanadium Oxide (VO2) M. Gatti et al, to be published Bandgap VO2 HF 7,6 DFT-LDA SC-COHSEX GW on SC-COHSEX 0 0,8 0,7 EXP 0,6

24 Vanadium Oxide (VO2) Need for self-consistency But static GW (COHSEX) self-consistent already ok! Bandgap VO2 HF 7,6 M. Gatti et al, to be published DFT-LDA GW on DFT SC-COHSEX GW on SC-COHSEX 0 0!!! 0,8 0,7 EXP 0,6

25 Quantum Transport and NEGF GW approximation and e-e scattering effects

26 Quantum Transport: The Working Bench V lead Left Contact µl lead conductor Right Contact µr V= L R Macroscopic Reservoirs: continuum of states, thermodynamic equilibrium Nanoscale Conductor: finite number of states, out of equilibrium, dissipative effects Mesoscopic Leads: large but finite number of states, partial equilibrium, ballistic We need: a First Principle description of the Electronic Structure for Finite Voltage: Open System and Out-of-Equilibrium description.

27 Non-Equilibrium Green's Function Theory (NEGF) (improperly called Keldysh) Much more complete framework, allows to deal with: Many-Body description of incoherent transport(electronelectron interaction, electronic correlations and also electronphonon); Out-of-Equilibrium situation; Access to Transient response (beyond Steady-State); Reduces to Landauer-Buttiker for coherent transport. The theory is due to the works of Schwinger, Baym, Kadanoff and Keldysh

28 Many-Body Finite-Temperature formalism T V W H= hamiltonian many-body o= i e Ei i o i i e e H = tr [e Ei H H ] o] =tr [ H observable statistical weight

29 NEGF formalism U t = V W U t t =H H T hamiltonian many-body + time-dependence o t ] o t =tr [ H H e H = tr [e H H ] t t 0 observable statistical weight referred to the unperturbed Hamiltonian and the equilibrium situation before t0

30 Time Contour o t = s t 0, t o t s t, t 0 Heisenberg representation H { t ' s t, t 0 =T exp i tt dt ' H 0 s t 0 i, t 0 =e H o t = } evolution operator trick to put the equilibrium weight into the evolution tr [s t0 i, t0 s t0, t o t s t, t0 ] tr [s t0 i, t0 ] o t = t ' o t ]] tr [TC [exp i C d t ' H t ' ]] tr [TC [exp i C d t ' H

31 NEGF Fundamental Kinetic Equations r r 1 G =[ H c ] G G r r =G =G G a G a a a Caveat!: in case we want to consider also the transient, then we should add another term to these equations: G r =G a r r G 1 G G 0 1 G Keldysh equation

32 Quantum Transport: composition of the Self-energy r = p r p interaction with the leads r e ph r e e electron-electron interaction ->? electron-phonon interaction -> SCBA (Frederiksen et al. PRL 2004) Critical point : Choice of relevant approximations for the Self-Energy and the in/out scattering functions

33 Our Self-Energy: GW. Why GW? Direct and Exchange terms: Band Structure Renormalization Selfconsistent Hartree Fock G0 W 0 Collisional Term: Band structure renormalization for Electronic Correlations + e-e Scattering -> Conductance Degrading Mechanisms, Resistance, non-coherent transport

34 GW and e-e scattering and correlation effects Gold Atomic Infinite Chain Loss of Conductance: Appearance of Resistance Appearance of Satellite satellite Conductance Channels Broadening of the peaks: QP lifetime P. Darancet et al, PRB 2007

35 C / V characteristics: GW vs EXP Gold Atomic Infinite Chain e-e e-ph } EXPERIMENT P. Darancet et al, PRB 2007

36 MBPT quantities as density-functionals: vertex corrections beyond GW

37 Hedin Equations G Σ Γ W Π Valerio Olevano, LEPES CNRS, Grenoble M M G M =1 =1 =1 GG Vc G Vc G

38 MBPT quantities as density-functionals: local vertex corrections beyond GW M M 1,2 ;3 =1 =1 Vc G M M 1,2 ;3 =1 =1 Vc M 1,2 G =1 G 5,7 G 6,8 7,8 ;3 Vc G 5,6 M 1,2 =1 4,3 Vc 4 Runge-Gross theorem Remainder Non-local Correction 1,2 ;3 =1 1,2 f eff xc 2,4 4,3 1,2 ;3 Direct gap LDA Si 2.53 Ar 8.18 GW Local Γ EXP F. Bruneval et al., PRL (2005) Direct gap COHSEX Si 3.64 Ar GW Local Γ EXP Valerio Olevano, LEPES CNRS, Grenoble

39 Generalized Sham-Schlüter Equation: link between non-locality and frequency dependence

40 Sham-Schlüter Equation KS G=G G KS V xc G Dyson Equation AND G KS x, x =G x, x = i r KS V xc = G 1 KS G G G Sham-Schlüter Equation, PRL (1983) 1 V xc = G G G G V EXX x 1 The density of the Kohn-Sham system is by construction equal to the exact density = G G G x G Linearised SSE Example: OEP EXact exchange

41 Generalize SSE Spectroscopy calls for the description of new quantities (ex. bandgap), beyond the ground-state density. SF I want the simpler one-body potential V able to provide the Green's function GSF of a Fictitious (Kohn-Sham-like) system such as by construction yields the exact density AND the exact photoemission bandgap. You can read the bandgap for example just only on the trace of the spectral function.

42 Generalized Sham-Schlüter Equation SF SF SF G=G G V G Dyson Equation AND SF G x, x =G x, x = i r The density of the SF system AND ℑ G SF is equal to the exact density r, r, = ℑ G r, r, = A r, r, The Trace of the spectral function is the exact one VSF r, = ℑ {GSF r, r1, G r1, r, } 1 ℑ{GSF r1, r2, r2, r 3, G r 3, r1, } Generalized SSE This is the real local and dynamical potential that yields the correct density and the correct bandgap!

43 Transforming non-locality Hartree-Fock self-energy on Jellium rs = 2.07 into frequency-dependence

44 Conclusions GW Quasiparticle band gaps and band plots are in good agreement with Photoemission spectroscopy. But the statistics is not yet quite large. We have still to see the role of selfconsistence and to which extent GW works on strongly-correlated systems. NEGF GW seems to introduce e-e scattering effects, correlation and lost-of-coherence in Quantum Transport. Setting MBPT quantities as density-functionals could be a good way to address vertex corrections beyond GW. Thank to Generalized SSE, we have introduced an effective framework which allows to get rid of the complicated non-local self-energy and have a simpler on-body local potential which yields the right bandgap. The effective potential is real but needs to be frequency-dependent. Valerio Olevano, LEPES CNRS, Grenoble

45 The ABINIT-GW code Σ = ig W in few words The thing: GW code in Frequency-Reciprocal space on a PW basis. Purpose: Quasiparticle Electronic Structure. Systems: Bulk, Surfaces, Clusters. Approximations: GW, Plasmon-Pole model and RPA on W, non Self-Consistent G0WRPA, first step of self-consistency on W and G. ABINIT is distributed Freeware and Open Source under the terms of the GNU General Public Licence (GPL). Copyright ABINIT GW group (R.W. Godby, L. Reining, G. Onida, V. Olevano, G.M. Rignanese, F. Bruneval)