Theory and Interpretation of Core-level Spectroscopies*

Transcription

1 Summer School: Electronic Structure Theory for Materials and Molecules IPAM Summer School, UCLA Los Angeles, CA 29 July, 2014 Theory and Interpretation of Core-level Spectroscopies* J. J. Rehr Department of Physics University of Washington, Seattle, WA Supported by DOE BES and NSF

2 Theory and Interpretation of Core-level Spectroscopies GOALS: ab initio theory of XAS, XPS, etc. Accuracy ~ experiment TALK: Excited state electronic structure I. Introduction History & motivation II. Real-space Green s Fn FEFF9 code III. GW/BSE OCEAN code IV. Extensions Correlated systems

3 I. Introduction: X-ray Absorption Spectra Theory vs expt EXAFS XAS fcc Al UV Photon energy (ev) X-ray Q: How to calculate & interpret?

4 Heuristic Interpretation of EXAFS* EXAFS Local structure Fourier Transform EXAFS Cu Shifted Radial Distribution shift X-ray Microscope! R nn *Stern Sayers Lytle, UW 1971 BUT need to calibrate experiment with Standard

5 Answer 1 I always thought it was easier to measure x-ray absorption than to calculate it. Hans Bethe ~ 1980

6 The chance is high that the truth lies in the fashionable direction. But on the off chance that it isn t, who will find it? R. P. Feynman

7 Gotcha: Standard DFT theory fails! Ground state = no damping LARGE ERRORS! Ground-state DFT theory (LAPW) vs Excited State vs Expt

8 Why the difficulty? Fermi Golden Rule for XAS μ(ω) Too many quasi-particle final states ψ f Final state rule V coul = V coul + V core hole Non- hermitian self-energy Σ(E) (replaces DFT Vxc)

9 Theoretical Breakthroughs Σ 4 developments beyond DFT + Scattering theory of excited states HIGH energy ~ 10 3 ev LARGE angular momentum l = 20 Inelastic mean free paths λ Screened core-hole Debye-waller damping factors

10 Answer 2 (JJR)* Now (1990) it is easier to calculate EXAFS than to measure it (**if structure is known) ** 10 years later - JACS 113, 5136 (1991)

11 How? Quantitative XAS theory FEFF J. J. Rehr & R.C. Albers Rev. Mod. Phys. 72, 621 (2000)

12 II. Real-space Green s Function Theory Golden rule via Wave Functions Ψ Paradigm shift Golden rule via Green s Functions G = 1/( E h Σ ) No sums over final states!

13 What s a real space Green s function? Wave function in QM H Ψ = E Ψ Ψ(r) = Amplitude to find particle at r Green s function (H E) G = - δ(r-r ) G(r,r,E) = aka Propagator = Amplitude to go from r to r Exact in various limits!

14 Multiple-scattering calculations of G MS Path Full MS Ingredients: G 0 free propagators t-matrix = e i δ l sin δ l δ RR δ ll

15 Implementation: RSGF Code FEFF 89 atom cluster Core-hole, SCF potentials Essential! BN

16 Example: Pt EXAFS MS path expansion Phase Corrected EXAFS Fourier Transform * Rnn= fcc Pt χ(r) Path Expansion 15 paths No peak shift! R (Å) *Theoretical phases accurate distances to < 0.01 Å

17 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 FEFF explains all spectral features, including absence of white line at L 2 -edge. Self-consistency essential 13360

18 Green s Functions and Parallel Computation Energy E is a parameter! 1/N CPU Natural parallelization Each CPU does one energy

19 Denstity of States interpretation of XANES : Excited State Electronic Structure Cu pdos vs XAS and XES XAS XES pdos Fermi energy E F Final state energy E

20 Application EXAFS analysis of Re-catalysts Simon R. Bare, et al., J. Phys. Chem. 115, 5740 (2011)

21 Application: Pt catalysts Pt 10 /γ-al 2 O 3 Finite Temp, Real-time DFT/MD + XAS of supported Pt nano-catalysts* Pt L 3 XAS fs time-steps Mean nn distance *F. Vila et al. Phys Rev B78, (R) (2008)

22 Experimental Observations (X-ray Absorption Expt)* 1st Neighbour Distance (Å) 2.85 Pt Foil 0.9-nm Pt on A (a) 2.70 NTE 0.9-nm Pt on A 2.4-nm Pt on C Enhanced σ 2 Normalized Absorption XAS Energy (ev) Red shift 165 K 200 K 293 K 423 K 573 K 2.65 (b) Temp FEFF explains all surprising anomalies *Kang, Menard, Frenkel, Nuzzo., JACS Commun. 128, (2006)

23 (10 more years) Improved many-body Corrections FEFF9 Theoretical Spectroscopy L. Reining (Ed) (2009) JJR et al., Comptes Rendus Physique 10, 548 (2009)

24 Key many-body effects in XAS Σ Self-energy Σ(E) Core-hole effects Multi-electron excitations Debye-Waller factors Energy shifts, losses Excitons, Screening Satellites Phonons, disorder

25 Many-pole GW Self-energy* MPSE: Σ(E) Energy dependent Extension of Hedin-Lundqvist plasmon-pole Sum of plasmon pole models matched to loss function LiF loss fn Efficient *J.J. Kas et. al, Phys Rev B 76, (2007)

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

27 Results: Accurate K-edge XANES Plasmon-pole self energy Cu Many-pole self energy

28 Ab initio XAS Debye Waller Factors e -2σ 2 k 2 * Ψ 2 β ω σ = ρ µ 0 ρ ( 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 *Phys. Rev. B 76, (2007)

29 Example: XAFS Debye-Waller Factor of Ge CD-Debye model LDA ab initio F. Vila et al. Expt: Dalba et al. (1999)

30 Many-body amplitude factors S 0 2 Q: What is S 0 2 S 02 ~0.9

31 Inelastic losses in XAS beyond QP Many-body XAS = Convolution of QP XAS with energy dependent spectral function A(ω, ω ) μ qp (ω) S 0 2 Explains crossover: adiabatic to sudden approximation

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

33 g eff (ω)= Effective GW++ Green s Function* Spectral function: A(ω) = -(1/π) Im g eff (ω) Extrinsic + Intrinsic - 2 x Interference Leading term: 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)

34 Implementation in FEFF Multi-electron excitations satellites in spectral funcion S 0 2 = 0.9 cf. W. Bardyszewski and L. Hedin, Physica Scripta 32, 439 (1985)

35 III. GW/Bethe-Salpeter Equation* 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

36 A. AI2NBSE code Optical-UV Spectra (ABINIT + NIST BSE + MPSE + interface): UW / NIST Collaboration Plane-wave, pseudopotential code cf. EXC, YAMBO, BerkeleyGW, etc.

37 B. OCEAN* Core-GW/BSE Code LiF: F K edge OCEAN vs FEFF *Obtaining Core Excitations from ABINIT and NBSE PW-PP + PAW + MPSE + NBSE Phys. Rev. B83, (2011)

38 Effect of Self-energy in BSE and ΔSCF-DFT* LiF (BSE < OCEAN) MgAl 2 O 4 < ΔSCF-DFT EXPT MPSE µ(e) (arb u.) MPSE 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)

39 Application: Water & Ice ( AI2NBSE + OCEAN ) < 17 molecule MD snapshots Optical Spectra O K-edge XAS

40 OCEAN L-edge Spectra Multiplet effects* Ti L 2,3 edge SrTiO3 J. Elec.Spect. Rel. Phen. 144, 1187(2005) a lá E. Shirley: BSE + KS crystal potential ab initio no parameters H BSE = H h + H e + H eh H h = ² + L S(p) H e = p2 2m + L S(d) + Hxtal KS H eh = V (r) + g(i; j) cf. De Groot et al Atomic multiplet model + crystal-field parameters H at = H av + L S(p) +L S(d) + H xtalfield + g(i; j)

41 IV. Extensions for correlated systems a) Hubbard-model corrections O K-edge MnO Hubbard U as self-energy correction estimated with crpa V U (r; E) = V SCF (r) + GW (E) + U lm¾ (E) cf. H. Jiang, Rinke et al. Phys. Rev. B 82, (2010).

42 Extensions b) Charge-Transfer Excitations* 3-Level system + Quasi-boson model Core c> *a lá Lee, Gunnarsson, & Hedin. (1999) Localized a> Ligand b>

43 Double pole intrinsic spectral function Ratio r(ω) ~ 6 ev

44 Example: CT satellites in transition metal oxides Physical Review B 89, (2014) Cf. J.D.Lee, O.Gunnarsson,and L.Hedin, Phys.Rev. B60,8034 (1999)

45 Extensions: RIXS and COMPTON Spectra

46 User-friendly code GUI interfaces JFEFF Proceedings of XAFS 15 in press

47 Summary & Conclusions Goals largely achieved - Core spectra understood in terms of excited state theory Ab initio RSGF and GW/BSE codes Ab initio many body corrections: core-hole, self energy, DW factors, and strong correlations Combined codes (FEFF+OCEAN+AI2NBSE) for full spectrum response UV-Vis-XAS

48 Acknowledgments: Rehr group & collaborators and L Reining, M Guzzo, E Shirley, K Gilmore, et al. Thanks to DOE BES and CMCSN and the ETSF for $ &