Core loss spectra (EELS, XAS)

Transcription

1 Core loss spectra (EELS, XAS) Kevin Jorissen University of Washington (USA) WIENk 013 Penn State

2 1. Concepts

3 WIENk calculates ELNES / XANES EELS : Electron Energy Loss Spectroscopy XAS: X-ray Absorption Spectroscopy Ionization edge ELNES EXELFS Energy Loss 3

4 the excitation process 4

5 INTRODUCING EELS Electron Energy Loss Spectroscopy is performed in a Transmission Electron Microscope, using a beam of high-energy electrons as a probe. The energy distribution of the beam gives a loss spectrum similar to XAS. Focussed probes give excellent spatial resolution (~0.5 Å). Energy resolution is improving ( ~ 5meV). Electron microscope equipped with EELS-detector Intuitive picture of EELS EELS spectra of TM oxides Probes local electronic structure 50.0%O- 50.0%Mn 66.7%O-33.3%Mn 55.0%O-45.0%Ti 66.7%O- 33.3%Ti 66.7%O- 33.3%Ti 5

6 Terminology for ionization edges Inner shell ionization. 6

7 instrumentation XAS: synchrotron EELS: microscope 7

8 THEORY OF EELS : A double differential scattering cross-section is calculated by summing over all possible transitions between initial and final states. The transition probabilities are described by Fermi s golden rule. k F ( E, Q) Ik V k F E E E k IF, V is the interaction potential between the fast beam electron and an electron in the sample. F, I the sample states, can be taken from electronic structure calculations. k F and k I the probe states, are typically described as plane waves when Bragg scattering effects are neglected. In experiment, one usually integrates over a range of scattering angles, due to the beam width and spectrometer aperture. differential cross section : k F ( E;, ) d Ik V k F E E E k, I IF, I F I F I I F I F 8

9 Theory (EELS XAS) E I,F I ei q. R F E I,F I ei q. R e RF 9

10 dipole approximation q. R 1 e i q R 1 iq. R ( q. R)! EELS EELS E I q. RF I,F XAS The polarization vector e in XAS plays the same role as momentum transfer q in ELNES within the dipole approximation. This is why people say XAS = EELS. (Beware - there are quite a few differences, too.) E I. RF I,F Probes local, symmetry-selected (l c +1) unoccupied DOS 10

11 . WIENk Calculations. 11

12 calculation of spectra using WIENk Set up structure and initialize SCF calculation x qtl -telnes Prepare case.innes x telnes3 or Prepare case.inxs x xspec EELS x broadening XAS 1

13 ELNES workflow 13

14 wweb ELNES input wweb 14

15 ELNES input file (case.innes) 15

16 16

17 Practical considerations Spectra usually converge easily with respect to RKMAX, k-mesh, SCF criteria But you should check anyway (see Cu L3) Optimizing positions may be necessary You may need to sum over all C atoms in the unit cell. (Especially for orientation-resolved calculations.) You probably need to use a core hole. This can be a lot of work. Your results may be wrong even if you do everything right. (But often they are reasonably good.) To compare to experiment, you ll probably fiddle with the broadening, the onset energy, and the branching ratio (L3/L) 17

18 Convergence of Cu L3 edge with # k-points 18

19 Features of WIENk Orientation dependence Beyond dipole selection rule Several broadening schemes All-electron For EELS: Account for collection/convergence angle Output (E) or () Relativistic ELNES ( anisotropic materials) 19

20 EELS Relativistic theory needed for anisotropic materials Semi-relativistic theory : 4 k ' 1 ( E, Q) I Q.r F EI EF E E a k Q 4 0 IF, V= r-r -1 m -> m E -> E,rel Fully relativistic theory (P. Schattschneider et al., Phys. Rev. B 005) : p.v 4e Up to leading order in c - and using the Lorentz gauge : k ' 1 ( E, Q) I. Q F E E E r Q e E k Q E / c IF, V e1 mc 0 z z I F q q.v 0 c Geometrical interpretation : in the dipole limit, a relativistic Hamiltonian shrinks the impuls transfer in the direction of propagation. (The general case is more complex.) WIENk can also calculate non-dipole relativistic transitions. The equations are so long they make PowerPoint cry. 0

21 Beyond the small q approximation The relativistic DDSCS : 4 a0 k f iqr. v. 0 p f e 1 i Ef Ei E E q ki i, f mec E c Dipole approximation : iv f i d Y t i t 5 iq.r f * * 4 lm q 1 a lm a q.v 0 i... f i r. q l v0 f z z1 ; x, y c l f * * i iv 0 4 ui mi idlm Y q juu l i ju l ui lm m m i mc 3 e r r More general l,m decomposition : f lm f dlm ul () r li 1 li 1 l li 1 li 1 li 1 l li 1 iv0 8 l 5 i mi li mi m f * * i miv i f 04 i md lm Y q t1 mi it m 0 a i mi m mi mc e 3 lm a q q q q unchanged 4 ju l ui r l l f * * i iv0 4 ui mi 4 idlm Y q juu l i ju l ui i i i i i i lm m mi mc e 3 r r li 1 li 1 l li 1 li 1 li 1 l li 1 iv0 8 mi m i 1 1m m i m ili miil i m1i 11m mi mi mi 0m mi mi mi 0m mi mc e 3 ui ju l r li 1 li 1 l li 1 li 1 li 1 l li 1 mi mi 1 1m mi mi mi 1 1m mi e 1iq.r l 1 l 1 l l 1 l 1 l 1 l l 1 1 1

22 Relativistic spectra Graphite C K for 3 tilt angles. Beam energy 300 kev, collection angle =.4mrad. Left: nonrelativistic calculation. Right: relativistic calculation.

23 Orientation dependence graphite C K EELS BN B K XAS 3

24 Spectrum as a function of energy loss E E d E,, f ( ) d k' kk' k k E Cr3C C K edge 4

25 Spectrum as a function of scattering angle e-05 1e-06 1e-07 1e-08 1e-09 1e-10 de E E1 E E q, q all transitions p -> s p -> p p -> d 1e scattering angle in mrad Left : L3 edge of Cr3C Right : the As L3 edge of NiAs (134 ev) Calculated using WIENk+TELNES 5

26 Just the double-differential CS Double differential scattering cross-section (DDSCS) E E, q 6

27 Warning! DFT is a ground state theory! it should fail for the prediction of excited state properties however: for many systems it works pretty well 7

28 The core hole E E E E F E F E F p s 1s p s 1s p s 1s ELECTRON MICROSCOPY FOR MATERIALS RESEARCH

29 Different ways of treating the core hole within WIENk No core hole (= ground state, sudden approximation) Z+1 approximation (eg., replace C by N) Remove 1 core electron, add 1 electron to conduction band Remove 1 core electron, add 1 electron as uniform background charge Fractional core hole: remove between 0 and 1 electron charge (e.g. 0.5) You may still get a bad result correct treatment requires a more advanced theory, e.g. BSE treats electron-hole interaction explicitly (gold standard). Core hole calculations usually require a supercell!!! 9

30 Mg-K in MgO mismatch between experiment and simulation introduction of core hole or Z+1 approximation does not help interaction between neighbouring core holes core hole in a supercell C. Hébert, J. Luitz, P. Schattschneider Micron 34, 19 (003) 30

31 Challenges of WIENk 1. Basis set only meant for limited energy range : forget about EXAFS/EXELFS sometimes adding a LO (case.in1) with a high linearization energy of.0 or 3.0 improves description of high-energy states.. Sometimes Final State Rule (core hole) DFT just isn t good enough and you need Bethe- Salpeter (BSE) calculations codes : OCEAN, AINBSE, Exc!ting, BSE much more expensive not as polished as DFT gets L3/L ratios right reality BSE single-particle 31

32 Challenges of WIENk 3. Core hole supercell size can be hard to converge. size of the cell how much charge to remove? optimal treatment can differ between similar materials; or even different edges in same material above: diamond GaN N K edge S. Lazar, C. Hébert, H. W. Zandbergen Ultramicroscopy 98, -4, 49 (004) 3

33 Challenges of WIENk 4. Killing artifacts (unphysical monopoles) by extending the RMT 33

34 Documentation WIENk Users Guide! C. Hebert, Practical aspects of calculating EELS using the WIENk code, Ultramicroscopy, 007 Jorissen, Hebert & Luitz, submitting ( thesis_jorissen.pdf - Kevin s Ph.D. thesis) 34

35 3. Hands-on exercises 1. XAS of K edge of Cu.. averaged EELS of N K edge of GaN. 3. orientation sensitive, in-plane and out-ofplane EELS of N K edge of GaN. 4. core hole calculation for Cu K-edge XAS & compare. 5. initialize a ** supercell for TiC or TiN core hole EELS calculation. 6. Be K edge. Find the error. 35

36 Thank you: C. Hebert, J. Luitz, P. Schattschneider, and the TELNES team P. Blaha, K. Schwarz, and the WIENk team J. Rehr and the FEFF9 team WIEN013 organizers 36