III. Inelastic losses and many-body effects in x-ray spectra

Transcription

1 TIMES Lecture Series SIMES-SLAC-Stanford March 2, 2017 III. Inelastic losses and many-body effects in x-ray spectra J. J. Rehr

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

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

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

6 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

7 Starting point for core-xas calculations: Golden Quasi-particle rule for XAS final via state Wave Green s Functions 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)

8 Many-pole GW Self-energy Σ(E)* Efficient GW approximation for Extrinsic Losses LiF loss fn - Im ε -1 Sum of plasmon-pole models matched to loss function Σ(E)= igw = Σ - i Γ W = ε -1 v Extension of Hedin-Lundqvist GW plasmon-pole model *J.J. Kas et. al, Phys Rev B 76, (2007)

9 Self-energy fixes systematic shifts & broadening 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)

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 PROBLEM: Amplitude discrepancy in EXAFS Observed fine structure smaller than QP theory S 0 2 ~ 0.9

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

14 Which Green s function? 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).

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

16 Reviews & references for cumulant Green s fn

17 Why does it work: Quasi-boson approximation IDEA: Neutral excitations - plasmons, phonons, etc. can be represented as bosons 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. Physics:** GW approximation describes an electronic-polaron: electrons coupled to density fluctuations modeled as bosons **B. I. Lundqvist, Phys. Kondens. Mater (1967)

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

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

21 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

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

23 Hedin suggestion: quasi-boson method with intrinsic, extrinsic and interference 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)

24 Starting point: GW/BSE 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

25 OCEAN: core-level 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)

26 Particle-hole cumulant in XPS* Europhys 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).

27 Quasi-boson method for particle-hole GF* 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)

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

29 Particle-hole cumulant in XAS* 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)

30 Theory of many-body amplitude factor 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!

31 Many-body amplitude factor S 0 2 MS Nano Proceedings, Springer (in press 2017)

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

33 Satellites and real-space interpretation RT TDDFT Cumulant Theory vs XPS Charge transfer fluctuations ω ct Interpretation: satellites arise from charge density fluctuations between ligand and metal at frequency ~ ω CT due to suddenly turned-on core-hole

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

35 Examples: high accuracy XPS and XAS Phys Rev B (in press, 2017) F. Fossard, K. Gilmore, G. Hug, J J. Kas, J J Rehr, E L Shirley and F D Vila RT-TDDFT intrinsic cumulant XPS Particle-hole cumulant

36 X-ray Edge Singularities in metals cf Doniach-Sunjic line-shape in XPS Low energy particle-hole excitations in cumulant explain edge singularities in XPS and XAS of metals Excitation spectrum

37 Correlated systems Standard approximation: Hubbard-model O K-edge MnO Hubbard U as self-energy correction V U (r; E) = V SCF (r) + GW (E) + U lm¾ (E) cf. H. Jiang, Rinke et al. Phys. Rev. B 82, (2010).

38 Alternative approach: cumulant Question: Does the particle-hole cumulant method work for correlated d- and f- 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)

39 Particle-hole cumulant for CeO 2 * Ce 5s XPS of CeO 2 Spectral function Ce L 3 XAS of CeO 2 Spectral weights *J. Kas et al. Phys Rev B 94, (2016)

40 Conclusions Many-body corrections including self-energy shifts, and inelastic losses, and Debye-Waller factors are essential for quantitative agreement with experimental x-ray spectra Particle-hole cumulant theory approximation can explain all losses (extrinsic, intrinsic and interference) in x-ray spectra. All losses can be lumped into a spectral function A K (ω) AND can be added ex post facto

41 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