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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).

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slide1

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).

slide2

Whatis XAS ?

Crystal

X-ray source

(synchrotron)

Incident X-ray

beam

TransmittedX-ray

Beam

Detector

slide3

XAS Physicalprocess

Challenge: describecore-hole attraction from first principles.

slide4

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

slide5

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/

slide6

Name of XAS structures

XANES

EXAFS

edge

pre-edge

XANES=X-raynearedge structures.

EXAFS=Extended X-Ray Absorption Fine Structure.

slide7

Name of XAS structures

XSPECTRA calculation

XANES

EXAFS

edge

pre-edge

XANES=X-raynearedge structures.

EXAFS=Extended X-Ray Absorption Fine Structure.

slide8

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.

slide9

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.

slide10

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

slide11

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)

slide12

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)

slide13

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)

slide14

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)

slide15

PAW for XAS – short explanation

By replacing

in the XAS matrixelement

Since the projectorssatisfy the condition,

weget:

M. Taillefumieret al. PRB 66, 195107 (2002)

slide16

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)

slide17

XAS Physicalprocess

Challenge: describecore-hole attraction from first principles.

slide18

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

slide19

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

slide20

Si K-edge in SiO2 ( alpha – Quartz)

[Si]=1s22s22p63s23p2

Excellent agreement Theory - Experiment

M. Taillefumieret al. PRB 66, 195107 (2002)

slide21

Core-holeeffects ?

(2x2x2)

(no core-hole)

Huge

core-hole

effects

M. Taillefumieret al. PRB 66, 195107 (2002)

slide22

Cu K-edge in bulkfcccopper

[Cu]=[Ar] 4s13d10

Moderatecore-holeeffects (almostcomplete screening)

C. Gougoussiset al. Phys. Rev. B 80, 075102 (2009)

slide23

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)

slide24

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?

slide25

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

slide26

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

slide27

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

slide28

Co K-edge XAS in LiCoO2

Competition betxeen core-hole attraction and electron-electron repulsion.

A. Juhinet al. Phys. Rev. B 81, 115115

slide29

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

slide30

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

slide31

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