Inelastic losses and satellites in x-ray and electron spectra*

Transcription

1 HoW Exciting! Workshop 2016 August 3-11, 2016 Humboldt-Universität -Berlin Berlin, Germany Inelastic losses and satellites in x-ray and electron spectra* J. J. Rehr, J. J. Kas & L. Reining+ Department of Physics, University of Washington, Seattle, WA, USA +CNRS, Ecole Polytechnique, Palaiseau, France *Supported by DOE BES DE-FG02-97ER45623

2 Inelastic losses and satellites in x-ray and electron spectra TALK: I. Introduction Many-body effects in XAS II. Inelastic losses & satellites III. Particle-hole theory: BSE Particle-hole cumulant Cumulant expansion beyond GW Intrinsic, extrinsic losses and interference

3 I. Introduction: Many-body effects in x-ray spectra Key many-body effects Core-hole effects Self-energy Σ(E) Phonons, disorder Excitations Excitonic effects, Screening Mean-free path, energy shifts Debye-Waller factors Inelastic losses & satellites

4 You can judge a many-body theory by how it treats the satellites. Lars Hedin (1995)

5 Quasi-particle theory of XAS Mini-review Theoretical Spectroscopy L. Reining, (Ed, 2009) JJR et al., Comptes Rendus Physique 10, 548 (2009)

6 Experiment: X-ray Absorption Spectra Theory vs Expt EXAFS fcc Al UV X-ray Photon energy (ev)

7 Motivation: Failure of ground-state DFT in XAS; need for inelastic losses Ground state No damping LARGE ERRORS! Ground-state DFT Excited State Expt NEED: energy dependent damping

8 Starting point for core-xas calculations: Quasi-particle final state Green s function Golden rule for XAS via Wave Functions Paradigm shift: Ψ Golden rule via Green s Functions G = 1/( E h Σ ) Final state h includes core-hole AND energy dependent self energy Σ(E)

9 Many-pole GW Self-energy Σ(E)* Extension of Hedin-Lundqvist GW plasmon-pole LiF loss fn Sum of plasmon-pole models matched to loss function Efficient GW method Σ(E)= Σ - i Γ *J.J. Kas et. al, Phys Rev B 76, (2007)

10 Core-hole potential - RPA W RPA Tungsten metal (Y. Takimoto) RPA a lá Stott-Zaremba Fully screened FEFF8 Unscreened cf. Screened core hole W in Bethe-Salpeter Eq Improves on final state rule, Z+1, half-core hole

11 Phonon effects: Debye Waller Factors in XAS e -2σ 2k2 * Ψ ρ 2 β ω σ = ρ µ 0 ( 2 ω ) coth dω i 2 ( 2 ) ( 2 ω Q δ ω D) = i Qi = { 6 step Lanczos recursion} Many pole model for phonons VDOS D dynamical matrix < ABINIT *Phys. Rev. B 76, (2007)

12 Self-energy largely fixes systematic errors due to self-energy in XAS DFT µ(e) (arb u.) GW- Self-energy MgAl 2 O 4 E (ev) *J. J. Kas, J. Vinson, N. Trcera, D. Cabaret, E. L. Shirley, and J. J. Rehr, Journal of Physics: Conference Series 190, (2009)

13 PROBLEM: Amplitude discrepancy - observed fine structure smaller than QP theory by factor S 02 ~0.9 - inelastic losses, multi-electron excitations S 02 ~0.9

14 Theoretical mysteries? Why does quasi-particle approx work well in XAS (~90%)?? Why are multi-electron excitations small in XAS (~10%)?? Why mysterious? Corrections to QP approx are large in electron gas Z ~ exp (-n)

15 II. Inelastic losses and satellites Q How to treat losses beyond the GW-quasi-particle approximaton? Approach: Improved treatment of G(E) including satellites in spectral function A(ω) = (1/π) Im G(E) Two methods: GW + Dyson Eq. Cumulant expansion

16 Which is better? GW + Dyson vs Cumulant* GW G(ω) = G 0 + G 0 Σ G Σ GW =igw Cumulant G(t) = G 0 (t) e C(t) C ~ Im Σ GW No vertex Γ = 1 Implicit vertex *Recent review and new derivation, see J. Zhou et al. J. Chem. Phys. 143, (2015).

17 Answer: from XPS Phys Rev Lett 77, 2268 (1996) Quasi-particle peaks of both GW and C agrees with XPS expt GW fails for satellites: only one satellite at wrong energy C: Cumulant model has multiple satellites ω p apart in agreement with expt Na XPS GW C C QP 2ω p ω p

18 Cumulant expansion properties* Excitation spectra (GW Σ) Spectral Function Landau formula for C(t) *For diagrammatic expansion of higher order terms, see e.g. O. Gunnarsson et al., Phys. Rev. B 50, (1994)

19 Example: Multiple Satellites in XPS of Si Si Quasiparticle peaks Multiple Satellites Lucia Reining Problems: GW: only one broad satellite at wrong position C: position ok but intensity too small

20 Quasi-boson approximation Theorem:* Cumulant representation of core-hole Green s function is EXACT for electrons coupled to bosons *D. C. Langreth, Phys. Rev. B 1, 471 (1970) Corollary: also valid for valence with recoil approximation. IDEA: Neutral excitations - plasmons, phonons, etc. can be represented as bosons Physics:** GW approximation describes an electronic-polaron: electrons coupled to density fluctuations modeled as bosons **B. I. Lundqvist, Phys. Kondens. Mater (1967)

21 Reviews/references for cumulant model

22 cf. Retarded Cumulant Approximation* Retarded GF formalism plasmaron Spectral function GC TO GW Builds in particle-hole symmetry

23 Electron-gas quasi-particle properties Retarded cumulant has good n k and Z, & pretty good correlation energies momentum distribution n k Z

24 Retarded cumulant for phonons* Spectral function cf A. Eiguren and C. Draxl, Phys. Rev. Lett. 101, (2008) Corrections to Migdal s theorem visible at low T

25 III. Particle-hole theory Q: How to calculate all inelastic losses and satellites in x-ray spectra? Single-particle cumulant in XPS (XAS) only has intrinsic (extrinsic) losses Extrinsic + Intrinsic - 2 x Interference

26 Suggestion from Hedin (1989): quasi-boson method for intrinsic, extrinsic, and interference terms Explanation of XAFS many-body amplitude factor:* χ exp = χ th * S 0 2 *J.J. Rehr, E.A. Stern, R.L. Martin, and E.R. Davidson, Phys. Rev. B 17,560 (1978)

27 GW/Bethe-Salpeter Equation* - Particle-hole Green s function w/o satellites Ingredients: Particle-Hole Hamiltonian H = h e - h h + V eh h e/h = ε nk + Σ nk Σ GW self-energy V eh = V x + W Particle-hole interaction

28 OCEAN* Core-GW/BSE Code LiF: F K edge Exp OCEAN FEFF9 *Obtaining Core Excitations from ABINIT and NBSE PW-PP + PAW + MPSE + NBSE *J. Vinson et al. Phys. Rev. B83, (2011)

29 Quasi-boson method for Particle-hole GF* Many-body Model: N> = e -,h, γ > V n -Im ε -1 (ω n,q n ) fluctuation potentials* Partition contributions into Intrinsic + Extrinsic + Interference * L. Hedin, J. Michiels, and J. Inglesfield, Phys. Rev. B 58, (1998)

30 cf Particle-hole Cumulant in XPS* Europhys J. J. B 85, 324 (2012) Kernel γ(ω) with extrinsic, intrinsic and interference terms *L. Hedin, J. Michiels, and J. Inglesfield, Phys. Rev. B 58, (1998).

31 Example: Satellites in XPS of Si again Si Multiple Satellites Quasiparticle peaks Success for particle-hole cumulant: good agreement when extrinsic and interference terms are included

32 Particle-hole cumulant for XAS* All losses in particle-hole spectral function A K NiO * cf. L. Campbell, L. Hedin, J. J. Rehr, and W. Bardyszewski, Phys. Rev. B 65, (2002)

33 Many-body amplitudes S 02 (ω) in XAS Many-body XAS Convolution μ qp (ω) S 02 (ω) Explains crossover: adiabatic S 02 (ω) = 1 to sudden transition S 02 (ω) 0.9 g q 2 = g q ext 2 + g q intrin 2-2 g q ext g q intrin Interference reduces loss!

34 Intrinsic losses: real-time TDDFT cumulant satellites Langreth cumulant in time-domain* TiO 2 *D. C. Langreth, Phys. Rev. B 1, 471 (1970)

35 Real-space interpretation: RT-TDDFT cumulant explains intrinsic excitations in TiO 2 RT TDDFT Cumulant Theory vs XPS Charge transfer fluctuations ω ct Interpretation: satellites arise from oscillatory charge density fluctuations between ligand and metal at frequency ~ ω CT due to turned-on core-hole

36 Extrinsic losses and Interference XAS of Al Satellite strengths Particle-hole cumulant explains cancellation of extrinsic and intrinsic losses at threshold and crossover: adiabatic to sudden approximation

37 X-ray Edge Singularities Low energy particle-hole excitations in cumulant explain edge singularities in XPS and XAS of metals Excitation spectrum

38 NIST Preprint; submitted to PRB 2016 F. Fossard, K. Gilmore, G. Hug, J J. Kas, J J Rehr, E L Shirley and F D Vila RT-TDDFT cumulant Particle-hole cumulant XPS

39 Question: Does the cumulant method work for correlated systems? Hedin s answer * MAYBE Calculation similar to core case but with more complicated fluctuation potentials V n -Im ε -1 (ω n,q n ) not question of principle, but of computational work... * L. Hedin, J. Phys.: Condens. Matter 11, R489 (1999)

40 Particle-hole cumulant for CeO 2 Ce 5s XPS of CeO 2 Spectral function Ce L 3 XAS of CeO 2 Spectral weights

41 Conclusions Particle-hole cumulant theory yields reasonable approximation for inelastic losses in XPS & XAS All losses (intrinsic, extrinsic and interference) in spectral function A K (ω) can be added ex post facto Interference terms explain mysteries in amplitudes and energy dependence: adiabatic- sudden transition Theory also applicable to some d- and f-systems.

42 Outlook Promising - much unexplored territory: Self-consistency, higher order cumulants? Quasi-particle properties, correlation energies? DMFT -. Casula et al, Phys. Rev. B 85, (2012) Finite temperature, magnons etc. etc. A lot more physics may be found by anyone daring enough to look for it. * *last sentence in Hedin s 1999 review

43 Acknowledgments: Supported by DOE BSE DE-FG02-97ER45623 Thanks to J.J. Kas L. Reining G. Bertsch J. Vinson K. Gilmore L. Campbell T. Fujikawa F. Vila E. Shirley S. Story S. Biermann M Guzzo M. Verstraete J. Sky Zhou C. Draxl et al. & especially the ETSF