Introduction to the calculation of X-ray absorption spectra

1. Introduction to the calculation of X-ray absorption spectra Matteo Calandra /afs/ictp/public/shared/qetutorial/Matteo/References/Lectures_XAS.pdf C. Gougoussis, M. Calandra, A. P. Seitsonen, and F. Mauri Phys. Rev. B 80, 075102 (2009) P. Giannozzi et al., J. Phys. Condens. Matter21, 395502 (2009).

2. Whatis XAS ? Crystal X-ray source (synchrotron) Incident X-ray beam TransmittedX-ray Beam Detector

3. XAS Physicalprocess Challenge: describecore-hole attraction from first principles.

4. Chemical and Orbital selectivity The choice of the incident photon energy selects the coreelectronexcited. It permits to choose (i) the atomfromwhich the electronisexcited (absorbingatom). (ii) the atomiclevelfromwhich the electronisexcited. Example: Eu L1, L2, L3edges Edgename Core state Intensity (Arb. Units.) Incident photon energy

5. Chemical and Orbital selectivity The choice of the incident photon energy selects whichcoreelectronisexcited. Experimentalists are then free to choose: (i) the atomfromwhich the electronisexcited (absorbingatom). (ii) the atomiclevelfromwhich the electronisexcited. Note thatcore-satebindingenergies of atoms are verydifferent: For more see http://xdb.lbl.gov/

6. Name of XAS structures XANES EXAFS edge pre-edge XANES=X-raynearedge structures. EXAFS=Extended X-Ray Absorption Fine Structure.

7. Name of XAS structures XSPECTRA calculation XANES EXAFS edge pre-edge XANES=X-raynearedge structures. EXAFS=Extended X-Ray Absorption Fine Structure.

8. XAS cross section From Fermi golden-rule: Final state with a core-hole Initial state: no core-hole Incident-photon energy The matrixelementis: 1s p states 1s d states See Ch. Brouder, J. Phys. Cond. Matt. 2, 701 (1990) for more details.

9. XAS cross section In the absence of a magneticfield the cross section canbeseparated in dipolar and quadrupolarterms (cross terms are zero in this case): with: and the twotermscanbecalculatedseparately. The quadrupolar part is in the pre-edgeregion.

10. XAS cross section needsall-electron states All-electron Final state with a core-hole Initial state: no core-hole Incident-photon energy ALL ELECTRON STATES

11. XAS cross section needsall-electron states Howeverwe have pseudopotentials and we do not have all-electron states! PSEUDO WAVE FUNCTION How to solvethisproblem ? Use the projectedaugmentedwave (PAW) method! P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)

12. PAW for XAS – short explanation As we use pseudopotentials, the electronwavefunctionthatweobtain in our simulation isdifferentfrom the all-electronwavefunction: ALL ELECTRON PSEUDO MAPPING Assume that a linearmappingexistsbetween the twowavefunctions: Nucleus As the pseudowavefunctiondiffersfrom the AE one only in the augmentation region (coreregion) then the mappingoperatoris differentfrom the identityonly in the augmentation region. Augmentation region P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)

13. PAW for XAS – short explanation The mappingiswritten as: R= coordinates of the nuclei with AE PARTIAL WAVES (e.g. AE atomic wavefunctions) PSEUDO PARTIAL WAVES (e.g. atomic pseudo Wavefunctions) Outside the augmentation region. P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)

14. PAW for XAS – short explanation The mappingiswritten as: R= coordinates of the nuclei with PAW Projectors are built to satisfyidentifiedthe conditions orthogonality Within the augmentation region. completeness P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)

15. PAW for XAS – short explanation By replacing in the XAS matrixelement Since the projectorssatisfy the condition, weget: M. Taillefumieret al. PRB 66, 195107 (2002)

16. PAW for XAS – short explanation islocalized on the absorbingatom site As the core state we drop all termswith Thus the all-electronmatrixelementiswrittenonly in terms of all-electronatomic partial waves, pawprojectors and the core initial state. Thusfrom an all-electroncalculation on the isolatedatomwecanget the state suchthat: M. Taillefumieret al. PRB 66, 195107 (2002)

17. XAS Physicalprocess Challenge: describecore-hole attraction from first principles.

18. XSPECTRA (Modeling the corehole) The core-holeismodeled in a supercellapproach. The pseudopotential of the absorbingatom has a core-hole in the core-state. As weworkwithperiodicboundarieswebuilda supercell to minimize the interaction between the core-hole and itsperiodicimages

19. How doesitworks in practice ? K-edge Weaklycorrelatedmaterials (SiO2, Cu, ….) Correlatedmaterials (NiO, Cuprates…) Large and correlatedsystems (Telephonenumber compound) L2,3-edges Copper, Cuprite K-edge XMCD Iron

20. Si K-edge in SiO2 ( alpha – Quartz) [Si]=1s22s22p63s23p2 Excellent agreement Theory - Experiment M. Taillefumieret al. PRB 66, 195107 (2002)

21. Core-holeeffects ? (2x2x2) (no core-hole) Huge core-hole effects M. Taillefumieret al. PRB 66, 195107 (2002)

22. Cu K-edge in bulkfcccopper [Cu]=[Ar] 4s13d10 Moderatecore-holeeffects (almostcomplete screening) C. Gougoussiset al. Phys. Rev. B 80, 075102 (2009)

23. Intersite 4p-3d hybridization NiO An example: NiO What atomic states form the dipolar part ? Ni atomic states 4.18 A Ni atomic 4p states are very extended: hybridization with off-site Ni 3d states is possible. Can 4p states promote transition to off-site Ni 3d states in dipolar Ni K-edge XAS ? Are these transition visibles ? T. Uozumi et al. Europhys. Lett 18,85 (1992)

24. Ni K-edge XAS in NiO - Experiments t2g : ε // [100] k // [010] eg : ε // [110] k // [-100] eg Open problems: - Excitations in the pre-edge and near-edge region, attribution unclear, A is considered quadrupolar in literature, all the others are clearly dipolar. - Can any of the dipolar feature be related to intersite 4p-3d hybridization?

25. Ni K-edge XAS in NiO - Theory Experiments: D. Heumann, G. Dräger, S. Bocharov - J. Phys. IV, 7 699 (1997) - U=7.6 eV (CALCULATED!) - Very good agreement up to the near-edge region - U seems to have no effect close to the edge. However in the pre-edge region… Ch. Gougoussisset al. Phys. Rev. B 79, 045118

26. Ni K-edge XAS in NiO - Theory t2g : ε // [100] k // [010] eg : ε // [110] k // [-100] - In DFT peak position and angular dependence are both incorrect. eg eg - DFT+U corrects the error. B A eg The use of U is mandatory to describe pre-edges of correlated magnetic insulators!!! eg A B t2g Ch. Gougoussisset al. Phys. Rev. B 79, 045118

27. Ni K-edge XAS in NiO - B peak interpretation core-hole effects included! The absorbing atom has up spin. - The B peak is due to transitions to up spin states, but the absorbing atom has 3d up states occupied! - The B peak is due to transitions to up spin states of next-to-nearest neighbouring Ni atoms. n. n. next to n. n. 1s hole Intersite Ni 4p-Ni 3d hybridization!!!! Ch. Gougoussisset al. Phys. Rev. B 79, 045118

28. Co K-edge XAS in LiCoO2 Competition betxeen core-hole attraction and electron-electron repulsion. A. Juhinet al. Phys. Rev. B 81, 115115

29. XAS in large systems – O K-edge in Telephone-number cuprates Chains • - 316 atoms AF unit cell • Correlationeffects (U) • Core-holeeffects Ladders Naturallydoped compound. Fondamental question: Where are holesitting, chain or ladders ? V. Ilakovacet al., Phys. Rev. B 85, 075108

30. XAS in large systems – O K-edge in Telephone-number cuprates C=Chain L=Ladder Legs R=LadderRungs Reduced gap = reduced distance betweenpeaks V. Ilakovacet al., Phys. Rev. B 85, 075108

31. XAS in large systems – O K-edge in Telephone-number cuprates Contrary to all previousanalysis, based on semi-empiricalmodels, U and antiferromagneticorderingfavor the strongchainhole-attraction. V. Ilakovacet al., Phys. Rev. B 85, 075108