Theory and Calculation of X-ray spectra. J. J. Kas

Transcription

1 Theory and Calculation of X-ray spectra J. J. Kas

2 Theoretical Spectroscopy Calculations GOAL: Next Generation Theory for Next Generation X-ray Sources TALK I Introduction II State-of-the-art III Next generation Current approximations Ab initio calculations Multi-electron excitations & strong correlations

3 What is x-ray absorption? photoelectron E E Fermi I = I 0 e -m(w )x Þ m(w)µ Log[I 0 / I] core level MSE-333 5/8/2013 3

4 What is x-ray absorption? L2,L3 L2,L3 L1 L1 μ(e) K K photoelectron E E Fermi core level MSE-333 5/8/2013 E (ev) 4

5 What is x-ray absorption? L2,L3 L2,L3 L1 L1 μ(e) K K photoelectron E E Fermi core level MSE-333 5/8/2013 MSE-333 5/8/ E (ev) 5

6 What is EXAFS? XANES EXAFS E) (Normalized Units) XANES - X-ray absorption Near Edge Structure EXAFS - Extended X-ray Absorption Fine Structure Cu K- edge 10 kev x- rays E Fermi X-ray energy E (ev) 6

7 What is EXAFS? Quantum interference MSE-333 5/8/2013 7

8 Qualitative Interpretation D.Sayers Stern & Lytle 1970 Short range order theory EXAFS χ(r) EXAFS Fourier transform radial distribution function Rnn= fcc Pt X-ray Microscope! R (Å) MSE-333 5/8/2013

9 Introduction Promise of Next generation Light sources: NSLSII, LCLS Powerful probes of multiple length and time scales Partial solution to DOE 5 Grand Challenges, etc. k GW/BSE True Crystals DFT/MD Molecular Crystals n IR V/UV X-Rays ns ps fs as RSGF Molecules Clusters Liquids RT-TDDFT r BUT Promise cannot be fully exploited without Next generation theory & software t

10 I Current methods EXAFS & XANES Automated & Integrated XAS codes FEFF > 20 yrs development J. J. Rehr & R.C. Albers Rev. Mod. Phys. 72, 621 (2000)

11 Real-space Green s Function (RSGF) Theory VIS X-ray Fermi golden rule via Wave Functions Ψ Paradigm shift: Golden rule via Green s Functions G = 1/( E h Σ ) EFFICIENT: No sums over final states!

12 Example: Pt XANES full multiple-scattering Pt L 3 -edge 1.4 Pt L 2 -edge (S. Bare, UOP) Normalized Absorption FEFF calculation Experiment' Normalized Absorption FEFF calculation Experiment PtL3_xmu '98feb002_xmu' PtL2_xmu '98feb004_xm 0.0 PtL3edge Photon Energy, ev PtL2edge Photon Energy, ev Good agreement: Relativistic FEFF8 code reproduces all spectral features, including absence of white line at L 2 -edge. Self-consistency essential: position of Fermi level strongly affects white line intensity

13 Spectra: RIXS, XES, Compton,

14 II. State of the art theory FEFF9 Theoretical Spectroscopy L. Reining (Ed) (2009) JJR et al., Comptes Rendus Physique 10, 548 (2009)

15 State of the art theory GOAL: ab initio many-body effects no adjustable parameters GW Self-energy Debye Waller factors RPA core-hole Need: multiple codes: DFT, self-energy, Phonons No one group or code can do it all

16 RPA Core-hole potential RPA Tungsten metal (Y. Takimoto) RPA Fully screened FEFF Unscreened * Also used in BSE Improves on final state rule, Z+1, half hole

17 Many-pole GW Self-energy Energy dependent Σ(E)=iGW replaces V xc MFP, energy shifts cf. Hedin-Lundqvist plasmon-pole Improved XANES LiF loss fn < AI2NBSE Cu J. Kas, UW *J.J. Kas et. al, Phys Rev B 76, (2007)

18 Ab initio XAS Debye Waller Factors * -2σ2 k2 e Ψ Many Pole model = 2 mi 0 w coth 2 w 2 w = Qi w - D Qi 2 dw 2 = 6 - step Lanczos recursion *Phys. Rev. B 76, (2007) for phonons VDOS

19 Example: XAFS Debye-Waller Factor of Ge F. Vila et al. Expt: Dalba et al. (1999)

20 Real-time Finite T Nanoscale Pt catalysts* Pt 10 / -Al 2 O 3 DFT-MD / XAS Expt. vs DFT/MD structure + FEFF XAS Theory* ~ 10 4 cpu-hr metallic Pt oxidized Pt Al O *F. Vila, J.J. Rehr, J. Kas, R.G. Nuzzo, A.I. Frenkel, Phys. Rev. B 78, (R) (2008)

21 EXAFS and DFT-MD Aqueous ions Transition metals, Me 2+ Exceptional agreement for: 1 Me 3+ st shell distance and disorder Octahedral symmetry Angular distribution functions Vibrational motions of first shell 2 nd shell distance and disorder Zn 2+ Radial Structure Plot, Img [ (R)], Å -3 MD- EXAFS QM/MM PBE0 with 3s and 3p QM/MM Cr H 2 O / 58 H 2 O DFT PBE0/CPMD MM SPC/E 300K Fulton, Bylaska, Schenter, et al JPC Lett., 3, 2588 (2012) 21

22 III. Next generation XAS Calculations - beyond-quasiparticles Excitonic effects BSE Multi-electron excitations Strong correlations Plus integrated Codes: DFT, GW/BSE, Real-time etc.

23 GW Bethe-Salpeter Equation BSE OCEAN* LiF: F K edge *Obtaining Core Excitations from ab initio NBSE Plane-wave, pseudo- potential + PAW + MPSE Phys. Rev. B83, (2011) cf. EXC, EXC!TING, BerkeleyGW

24 Many-body Amplitudes in XAS S 0 2 Many-body X*S Convolution Source: multi-electron excitations Na XPS Spectral function ~ XPS A = - Im G GW C

25 Which Green s function? GW vs Cumulant Σ = i GWΓ GW G(ω) = G 0 + G 0 Σ G Cumulant G(t) = G 0 (t) e C(t) Γ = 1 Σ GW =igw C ~ Im Σ GW Similar ingredients; all many-body effects in Σ GW

26 Test: Satellites in XPS of simple metals* Phys Rev Lett 77, 2268 (1996) Good news: GW quasi-particle peak agrees with XPS Bad news: GW has one wrong energy XPS has multiple satellites GW Na XPS QP

27 Test 2: Multiple Satellites in XPS of Si Multiple Satellites Lucia Reining Quasiparticle peaks

28 XPS Intensity Ratios in Molecules X = EXAFS <XPS> = XAS Intensity ratio ~ Z FEFF9 including all losses explains nonstoichiometric ratios

29 Question: Can the cumulant method work for CT satellites in correlated systems? Hedin s answer * MAYBE Calculation similar to core case but with more complicated fluctuation potentials not question of principle, but of computational work... * L. Hedin, J. Phys.: Condens. Matter 11, R489 (1999)

30 Approach: Cumulant expansion in real-time arxiv: Langreth RT Cumulant: RT-TDDFT

31 XPS & Real-space Interpretation RT TDDFT Cumulant Theory vs XPS Charge Fluctuations ω ct Interpretation: satellites due to local ligand-metal charge fluctuations at frequency ~ ω ct in response to transient core-hole potential JJ Kas, FD Vila, JJR & S.Chambers arxiv:

32 Conclusions and Outlook EXAFS well understood with current methods RSGF + DFT/MD XANES semi-quantitative state of the art theory GW, GW/BSE, DW factors, XAS, XPS Multi-electron excitations & correlations via quasi-boson/cumulant theory Full spectrum calculations possible in arbitrary materials with NG theory and codes.

33 Rehr group & collaborators Thanks for Listening!

34 Many Success Stories XAS of doped SiC M Ohkubo et al., Sci. Rep., 2, 831 (2012) FEFF8.4 played an important role in analysis of XANES spectra of N-doped SiC. I appreciate your effort for developing this fantastic software Masataka Ohkubo

35 Cumulant Expansion & Spectral Function Spectral Function *see L. Hedin, J. Phys.: Condens. Matter 11, R489 (1999) cf. D.C. Langreth Phys. Rev. B 1, 471 (1970) (Linked cluster theorem)

36 Other properties Kernel: β(ω) = Im Σ k (ω) ~ Im ε -1 ~ loss function Peaks in A(ω) ~ Peaks in loss function mean number of shake-up bosons Dimensionless measure of correlation strength n ~ 0.3 to 0.4 even in s-p systems!

37 IV. Problems with Cumulant Expansion* BAD: Pathologies at Fermi energy* RC TO GW Electron spectral function Time-ordered GF (TO) lacks satellites on both sides of QP peak near Fermi energy GW lacks multiple satellites direct exchange Source: recoil approximation: neglect of exchange diagrams in time-ordered cumulant *arxiv:

38 Debye-Waller factors σ 2 in Zr tungstate

39 FIX: Retarded Cumulant expansion* Retarded formalism Spectral function GC TO GW *Phys. Rev. B. (in press, Aug. 2014); arxiv:

40 Retarded Cumulant (GC) gives GOOD quasi-particle properties momentum distribution n k Z

41 Many-body Amplitudes in X-ray Spectra Many-body X*S Convolution Explains crossover: adiabatic to sudden transition g q 2 = g q ext 2 + g q intrin 2-2 g q ext g q intrin Interference reduces loss!

42 Strong correlation effects: 1. Hubbard corrections crpa calculated U

43 FIX: Next generation approach Quasi-Boson Approximation IDEA: Neutral Excitations - plasmons, phonons, * electron-hole pairs, magnons are bosons γ Many-body Model: N> = e -,h, γ > V n -Im ε -1 (ω n,q n ) fluctuation potentials * L. Hedin, J. Phys.: Condens. Matter 11, R489 (1999)

44 g eff (ω)= GW++: Effective Green s Function - perturbation theory* Extrinsic + Intrinsic - 2 x Interference Leading term: g (ω) damped Green s function in presence of core-hole (~ final state rule! ) *L. Campbell, L. Hedin, J. J. Rehr, and W. Bardyszewski, Phys. Rev. B 65, (2002)

45 2. Multiplets a la GW/BSE Phys. Rev. B 86, (2012) GW/BSE + Spin & SO + MPSE + Slater F, G Transition metal oxides Multiplets in SrTiO 3

46 3. Charge Transfer Satellites 2-state picture Strong Screening 3d 10 4s 1 Weak Screening 3d 9 4s 2 Approaches: Anderson impurity, DMFT, multiplets,. + Lots of Sound & Fury + millions of cpu hrs BUT still no satisfactory 1 st principles theory

47 Simple picture: 2-state model of LGH* H = n U n n t( c c c c ) i DFT i i ij i j 3d 4s 4s 3d ij XAS NiO ~ 7 ev *Lee, Gunnarsson & Hedin, Phys Rev B 60, 8034 (1999)

48 Goal: X-ray Spectroscopy Beamline Full spectrum theoretical tools XAS XES XMCD NRIXS RIXS etc. UV XRAY

49 UW EU Model European Theoretical Spectroscopy Facility

50 Networked theoretical beamlines