1. This work was funded in part by the US Department of Energy Grant No. DE-FG03-97ER45623 through the Computational Materials Science Network (CMSN) and the Office of Microelectronics Programs at NIST and International SEMATECH. Theoretical Modeling of Core Excitation Spectra Eric L. Shirley (NIST),* on behalf of *Tel: 301 975 2349 / FAX: 301 975 2950 / email: eric.shirley@nist.gov L.X. Benedict, LLNL J.A. Soininen, U. Helsinki Z.H. Levine, NIST J.A. Burnett, NIST J.J. Rehr, U. Wash., Seattle S. Dalosto, NIST H.M. Lawler, U. Wash., Seattle E.K. Chang J.A. Soininen, U. Helsinki J.J. Rehr, U. Wash., Seattle J.C. Woicik (NIST) C.S. Hellberg (NRL), E.K. Chang H.M. Lawler, U. Wash., Seattle … and others! (NIR/VIS/UV) (x-ray) (Thz)

2. Index of refraction n and index of absorption k in GaAs: [taken from Palik, Handbook of Optical Constants of Solids, Volume I.] TO phonon 2-phonon... 3 phonon x-ray edges Optical constants through the electromagnetic spectrum: Various excitations in a solid and associated excitation spectra: inter-band transitions VIS/UV optical properties core excitations lattice vibrations (phonons) 2-phonon

3. Definition of terms: If dielectric tensor may be treated as a scalar, one has Some related quantities: Note: Atomic units

4. TALK PLAN: Against the backdrop of the multi-faceted interaction of light and matter, especially in solids, we shall discuss now-common model (GW/Bethe-Salpeter, a.k.a. GW/BSE) to calculate effect of electron-electron/electron-hole interactions on optical constants. This model includes: (1) self-consistent density-functional calculations to get good one-el. orbitals and energy bands, (2) many-body corrections to band energies (here, GW approximation), (3) a method to solve the coupled el.-hole e.o.m. describe the excited states resulting from excitation of an electron across the Fermi level. Sample results include: * calculation of absorption spectra involving core excitations * treatment of IXS at a a core edge (F 1s in LiF) * effects of electron/hole lifetime damping on core/valence spectra * quadrupolar x-ray transitions in transition metal (TM) oxides * multiplet effects in TM oxides * structural determination of ferroelectric TM oxide thin-films

5. Bethe-Salpeter calculations, including vs TD-DFT G. Onida, L. Reining, A. Rubio Electronic excitations: density-functional vs many-body Green's functions approaches Rev. Mod. Phys., 2002 O. Pulci, M. Marsili, E. Luppi, C. Hogan, V. Garbuio, F. Sottile, R. Magri, and R. Del Sole Electronic excitations in solids: Density functional and Green's functions theory Phys. Stat. Sol. (b), 2005 M. Palummo, O. Pulci, R. Del Sole, A. Marini, P. Hahn, W.G. Schmidt, and F. Bechstedt The BSE: a first-principles approach for calculating surface optical spectra J. Phys.: Condens. Matter, 2004 GW (and some GW/BSE) calculations F. Aryasetiawan and O. Gunnarsson The GW method Rep. Prog. Phys, 1998 L. Hedin On correlation effects in electron spectroscopies and the GW approximation J. Phys.: Condens. Matter, 1999 W.G. Aulbur, L. Jönsson, and J.W Wilkins Quasiparticle Calculations in Solids Solid State Physics, 2000 XAFS J.J. Rehr and R.C. Albers Theoretical approaches to x-ray absorption fine structure Rev. Mod. Phys., 2000 Classic L. Hedin and S. Lundqvist Effects of Electron-Electron and Electron-Phonon Interactions on the 1-el. States of Solids Solid State Physics, 1969 Review Articles:

6. Describing electron states Dyson equation: GW self-energy (accounts for many-body electron-electron interaction effects) electron state energy, wave function Predictive theory needs: * accurate band structure methods (Schrödinger equation) * many-body corrections to band energies GW LDA this work: pseudopotential, plane-wave, + PAW-type reconstruction as needed Energy/momentum domain: self-energy = convolution of electron propagator, polarization effects  electron interacts with its own polarization cloud Electron self-energy (Hedin ’ s GW approximation)

7. Excited state= linear superposition of all states produced by a single electron excitation. In each such electron-hole pair state, electron in band n, with crystal momentum k+q. hole in [ band or core-level ] n, with crystal momentum k, Call such a state |n n k(q) , total crystal momentum q. momentum E el Bethe-Salpeter calculation:

8. In a non-interacting picture, one has [ H  E 0 ] |n n k(q)  = [ E el ( n, k+q)  E el ( n, k) ] |n n k(q) . The e-h pair states {|n n k(q)  } diagonalize Hamiltonian, H. In an interacting picture, one has [ H  E 0 ] |n n k(q)  = [ E el ( n, k+q)  E el ( n, k) ] |n n k(q)  +  n  n  k V(n  n  k, nn k) |n  n  k(q) . This couples {|n n k(q)  } states. Stationary states are linear combinations of e-h pair states. BSE, cont’d.: Resulting coupled, el.-hole-pair Schrödinger equation of motion, i.e., Bethe-Salpeter equation: difficult to solve, especially within a realistic treatment of a solid. WARNING: This Bethe-Salpeter treatment is good when a one-electron/one-hole description is valid, and not otherwise. Maps onto 1-el. eq ’ n in cases of core holes without dynamics (e.g., 1s) H=H e +H h +H eh H=He+HhH=He+Hh

9. BSE, cont’d.: Two parts of electron-hole interaction: Direct: Exchange: The attractive direct part is responsible for excitons (bound electron-hole pairs) and an energetically downward shift in oscillator strength (excitonic effects). The repulsive exchange part is responsible for plasmons. Including it accounts for spin degrees of freedom for spin-singlet excitons.

10. MgO optical constants:

11. Li halides, Li 1s excitations (LiH, LiF, LiCl, LiBr, LiI, LiAt*) J. El. Spect. Rel. Phenom. 137-40: 579 (2004) Trends on descending through series, effects of electron-hole interaction *unstable measured spectrum (meas.) calculated spectrum (inter.) calculated spectrum, omitting electron-hole interaction (n.i.)

12. Dynamic structure factor for LiF at fluorine K edge (x-ray scattering): Momentum-dependent spectra betray even-parity core-hole exciton level. Calc. Meas. Data collected at European Synchrotron Radiation Facility (ESRF) q,  q in, E in q out, E out q = q in  q out,  = E in  E out. Scattering process: Hämäläinen et al., PRB 65, 155111 (2002).

13. Electron self-energy for band states in LiF, versus LDA band energy. (Self-energy minus average LDA exchange and correlation shown; conduction band minimum defined as zero.) F 2s F 2p Damping of electron states LDA=independent electron picture, without effectual self-energy effects

14. F 1s edge results in LiF with and without core-hole, damping. Damping/self-energy shifts realized by probing e-h Greens function, not at E, but at E+  (E); outstanding issue: what is self-energy of e-h pair?  VcVc core-hole effect damping effects electron-core hole Bethe-Salpeter equation Damping of electron/core states Effective 1-el. e.o.m. For el-core hole pair:

15. What about lifetime effects in valence spectra? Idea: average lifetime damping of electron and hole states at half the photon energy away from the mid-gap. needs lifetime broadening?

16. Quadrupole (and dipole) core-excitation spectra: “Pre-edge” and near-edge excitations in rutile TiO 2 XAS J. El. Spect. Rel. Phenom. 136, 77 (2004) dipole quadrupole Ti(1s  3d) Ti(1s  4p), etc. why does the spectrum worsen above here? damping eff. Rutile TiO 2 crystal structure radial integrals angular integrals Atomic-basis expansion of Bloch function: note possibility of longit. quadrupole in EELS/IXS

17. Multiplet effects in 3d L 2,3 oxide spectra Ti 4+ (1s 2 2s 2 2p 6 3s 2 3p 6 )  Ti 4+ (1s 2 2s 2 2p 5 3s 2 3p 6 3d 1 ) * 2p core hole M L,M S degrees of freedom (DOF) - exchange interaction with electron - multipole effects - spin-orbit effects * excited electron spin DOF - exchange interaction with hole - spin-orbit effects * particles interact - Coulomb monopole interaction - Multipole exchange & direct - Screening by other particles (  0.83) - Charge-transfer (beyond BSE) * solid-state environment - crystal field Chicken-egg paradigm: Should we view this as * crystal field effects in atomic calculation or * fancier excitonic effects in BSE calculation? Atomic view of Ti L 2,3 edge in, say, SrTiO 3 (taken from de Groot et al., PRB, 1990): ISOLATED ATOM ATOM + CRYSTAL FIELD WHICH CAME FIRST?

18. spin-orbit splitting crystal-field splitting spin-orbit and crystal-field splittings evident re-arranged oscillator strength Multiplet effects in 3d-oxide L 2,3 spectra: Example: Ti L 2,3 in SrTiO 3 J. El. Spect. 144, 1187 (2005) band-induced width; higher-lying features included naturally; C-T satellites absent This work: Bethe-Salpeter solid-state calculation: central part screened by RPA Slater-type integrals, scaled by  0.83 Generalize:

19. wide band gap (~12 eV) small band gap (~3 eV) Quantifying effects of correlation/multiple excitation beyond current Bethe-Salpeter treatment Results suggest that Coster-Kronig, vibrations shake-up, charge transfer excitations, etc. dominate remaining discrepancy (cf. de Groot et al.).

20. Thin-film strontium titanate on Si (100) Coherent growth of 5 ML film: compressive strain in a-b plane, c/a stretch, Leading to AFD cage rot. & FE polarization Local atomic geometric in bulk (a) and film (b) Ti K and L 2,3 spectra (expt=top, theory=bottom) BULK FILM

21. Electron damping (picture from Hedin Lundqvist 1969 Solid State Physics Review) Gradual onset from low-energy electron-hole (inter-band) continuum Main onset and high-energy tail from plasmon + large-q transitions Plasmon-pole model misses electron-hole continuum Example: bromellite Dielectric function betrays inter-band transitions that give lowest structure in loss function. tail broadens plasmon loss 05/09

22. Observed requirements of dielectric function: 1.) accounting for electron-hole continuum 2.) broadening of plasmon (high-energy tail in  2 )  Need to go beyond plasmon-pole model! 06/09 2(q,E)2(q,E) E E0E0 (q-dependence of C, ,  is implicit) Proposed model for dielectric function: Requires: approximate band gap E 0,  , occ ’ d band states C, ,  are known We obtain M 1 (q) from the f-sum rule M  1 (q) from the Levine-Louie dielectric function (plus Kramers-Kronig) M 0 (q) from

23. Other results (periclase loss function, silicon dielectric function): Sample results: Periclase small-q dielectric function 08/09

24. Effect on electron self-energy using GW approximation with jellium G, model W (screened interaction), which is an easy calculation. Inset: improved “ ramp-up ” of lifetime damping because of spread of oscillator strength in response function. Magnitude of imaginary part of self-energy: points=numerical calc (hours) curves=our model (seconds) Real part of self-energy: points=numerical calc (hours) curves=our model (seconds) Typical plasmon-pole model result Electron state energy minus conduction band minimum (eV)

25. n, k Silicon optical constants: x-ray to Thz Notes: NBSE denotes the NIST Bethse-Salpeter Equation (BSE) code(s); at.=first-principles correction for outer shell absorption asymptote; For DFPT results, see, for instance, Deinzer and Strauch, Phys. Rev. B, 2004 So where are we as a community over the whole spectrum? At least this good … (measured=Palik Handbook; Ikezawa and Ishigame, J. Phys. Soc. Japan, 1981) K L1L1 L 2,3 valence excitations differencessums phonons

26. Closing Remarks -Present BSE is a robust method to calculate many excitation spectra -Vibrational effects being included: work underway -Stronger correlation effects (greater entanglements in (1) ground-state wave function, (2) excited state wave function beyond el-hole pair -Generic “bridge” coupling valence BSE code to output of many plane-wave/pseudopotential codes is in place -“bridge” coupling core BSE code to output of many plane-wave/pseudopotentials codes conceived -with such bridge, codes would be shared with others