Optical properties, Core-level spectroscopy, GW and BSE

Optical properties, Core-level spectroscopy, GW and BSE

Peter BlahaInstitute of Materials Chemistry

TU WienXavier Rocquefelte

University of Rennes

1 – MULTIPLE FACETS OF COLOURED MATTER

2 – ELECTRONIC STRUCTURE OF A SOLID

3 – UNDERSTANDING OF COLORS FROM BANDS

4 – LIGHT‐MATTER INTERACTION

5 – OPTICAL PROPERTIES: WHICH TREATMENT?

6 – ILLUSTRATIONS

Modelling the Optical Properties of Inorganic Materials


DiffusionAbsorption

Coloring and pigments

Sky and clouds Others diffusingSystems …

Intermediate cases

Photons in interactionwith matter

Light‐Matter interaction

Zone ofinterferences

RefractioncolorsRainbow

Diffractioncolors

Interferentialcolor

Waves intransparentmedium

DiffusionAbsorptionLight‐Matter interaction



Intermediate cases



Zone ofinterferences

RefractioncolorsRainbow

Diffractioncolors

Interferentialcolor

Waves intransparentmedium

DiffusionAbsorptionLight‐Matter interaction



Intermediate cases


Eye

Brain Colored shadowCriteria colorsharmony

Colorimetry Color mixing

Light‐Matter interactionRetinal

in interactionwith photons

Nervoussignal


Interaction between the electromagnetic field and matter

Physical color

Diffusion et interferencesDispersion / Reflection et Refraction / Scattering

Transparent matter


Interaction between the electromagnetic field and matter

Physical color Chemical color

Diffusion et interferencesDispersion / Reflection et Refraction / Scattering

AbsorptionEnergy dissipation

Transparent matter Opaque matter


Chemical colors

AbsorptionEnergy dissipation

Opaque Matter

Acceptor electron levels Dissipative absorption

h

VB

CB

Gap

VB

CB

VB

CBUV

IR0

E (eV)

1.65

3.26

Chemical color « Inelastic diffusion »

From insulator to semiconductor to metal systems



p

s

From the atom to the molecule and to the solidEnergy

AtomAtomicOrbitals

p

s

*LCAO : Linear Combination of Atomic Orbitals

MoleculeLCAO*

MolecularOrbitals

From the atom to the molecule and to the solid

AtomAtomicOrbitals

Energy


SolidDispersion

Eg

CB

BV

ConductionBand

ValenceBand

BandGap

Bloch Orbitals= Bands

p

s

*LCAO : Linear Combination of Atomic Orbitals

MoleculeLCAO*

MolecularOrbitals

AtomAtomicOrbitals

Energy

From the atom to the molecule and to the solid


h

BV

CB

UV

IR

E (eV)

0

1

2

3

4

5

6

h

VB

UV

IR

E (eV)

0

1

2

3

4

5

6 Color of a SC* with 1 gap

*SC : Semiconductor


GAP

Abso

rptio

n (a

.u.)

Energy (eV)0 1 2 3 4 5 6

Transmittedlight

Color of thecompound

BV

CB

BV

CB

VB VB

Gap

0 1 2 3 4 5 6

hUV

IR

E (eV)

0

1

2

3

4

5

6

hUV

IR

E (eV)

0

1

2

3

4

5

6


Color of a SC* with 1 gap

Abso

rptio

n (a

.u.)

Energy (eV)

Transmittedlight

*SC : Semiconductor


BV

CB

BV

CB

BV

CB

VB VB VB

BI

Abso

rptio

n (a

.u.)

Energy (eV)0 1 2 3 4 5 6

No transmittedlight

*SC : Semiconductor

hUV

IR

E (eV)

0

1

2

3

4

5

6

hUV

IR

E (eV)

0

1

2

3

4

5

6


Color of a SC* with 1 gap


0 1 2 3 4 5 6BV

CB

VB

TransmittedLight

Energy (eV)

Abso

rptio

n (a

.u.)

EnergyColor of a SC* with several gaps

*SC : Semiconductor


*SC : Semi‐conducteur – BI : Bande interdite

0 1 2 3 4 5 6Energy (eV)

Abso

rptio

n (a

.u.)Transmitted

Light

BV

CB

VB

EnergyColor of a SC* with several gaps


Powder sample of TiO2 (rutile)

Charge transfercolor

TiO2


‐10

0

10

Energy (eV)

‐5

5

2 O2‐ (2p6)

Ti4+ (3d0)

“t2g”

“eg”

Eg 3 eV

TiO2

O Ti

Powder sample of TiO2 (rutile)



‐10

0

10

Energy (eV)

‐5

5

2 O2‐ (2p6)

Ti4+ (3d0)

“t2g”

“eg”

Eg 3 eV

TiO2

O Ti

Absorption (a.u.)

Energy (eV)0 1 2 3 4 5 6

UV



‐10

0

10

Energy (eV)

‐5

5

2 O2‐ (2p6)

Ti4+ (3d0)

“t2g”

“eg”

h> 3 eV

TiO2

Absorption (a.u.)

Energy (eV)0 1 2 3 4 5 6

UV

O Ti



TiCl3in solution

6 O2‐ (2p6)

Ti3+ (3d1)“t2g”

“eg”

h> 3 eV

h 2,5 eV

Ti[H2O]3+

d‐d transition color


Ti[H2O]3+

6 O2‐ (2p6)

Ti4+ (3d1)“t2g”

“eg”

h> 3 eV

h 2,5 eV

Absorption (a.u.)

Energy (eV)0 1 2 3 4 5 6

UV

Rq. : For dn with n > 1 Interelectronic correlations must be taken into accountTanabe‐Sugano diagrams(spectroscopic terms)

d‐d transition color


Aluminecorindon(Al2O3)

BV

CB

VB

EnergieAbsorption (a.u.)

Energy (eV)0 1 2 3 4 9

UV

10 11 12


Aluminecorindon(Al2O3)

Syntheticrubis

(Al2O3:1%Cr3+)

+ 1% Cr3+

Absorption (a.u.)

Energy (eV)0 1 2 3 4 9

UV

10 11

Absorption (a.u.)

Energy (eV)0 1 2 3 4


Impurity (d‐d) color

Rubis (Al2O3:1%Cr3+)

Cr3+ : electronic conf. d3 Interelectronic correlations must be taken into accountTanabe‐Sugano diagrams(spectroscopic terms)


Multiplicity of spin

Cr3+ : elect. conf. d3

2S+1X

Energy

4F0

4P

2G

2F

No CF*


Free ion: we define the spectroscopic terms :

4Fml +2 +1 0 ‐1 ‐2

L = +3

S = 3/2

*CF : Cristal Field



Energy

4F0

4P

2G

2F

2E2T1

2T2

4A2 (t23e0)

4T2 (t22e1)

4T1 (t21e2)

2A1

No CF* With CF*


Degeneracy lifting due to the octahedralcrystal field effect created by the ligands

Cr3+ : conf. eq d3

t23

e0

t22

e1

t21

e2

Excited States

Fundamental state


*CF : Cristal Field


Energy

4F0

4P

2G

2F

2E2T1

2T2

2A1

No CF* With CF**CC : Champ Cristallin


Energy (eV)0 1 2 3 4 5

Red color+

Slightly blue

Abs. (a.u.)

4A2 (t23e0)

4T2 (t22e1)

4T1 (t21e2)


Cr3+ : electronic conf. d3 Interelectronic correlations must be taken into accountTanabe‐Sugano diagrams(spectroscopic terms)


nvac. = 1

v = c/n

c

Example of a prism

Physical color « Elastic diffusion »

n : index of refraction

Propagation speed is changed


nvac. = 1

v = c/n

c

Elastic diffusion mechanismat the atomic level

Succession of absorption and emission

: Inductive dipole moment Receiver‐transmitter antenna



Example of a prism

n : index of refraction

Propagation speed is changed

Example of a prism

Oscillating field

Without applied field

Electronic density

re .Dipole moment

nvac. = 1

v = c/n

c


Elastic diffusion mechanismat the electronic level

Succession of absorption and emission

: Inductive dipole moment Receiver‐transmitter antenna

Oscillating Electric Field polarize


0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)0 5 10 15 20

0.0

0.5

1.0

1.5

Energy (eV)

n k

Diffusion Absorption

Kramers‐Kronig

Complex index of refraction: N = n + i k


0 5 10 15 200.0

1.0

2.0

3.0

4.0

Energy (eV)0 5 10 15 20

0.0

1.0

2.0

1.5

4.0

Energy (eV)

1 2Complex dielectric function: = 1 + i 2

1 = n2 – k2

2 = 2nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)0 5 10 15 20

0.0

0.5

1.0

1.5

Energy (eV)

n k

Diffusion Absorption

Kramers‐Kronig

Complex index of refraction: N = n + i k


0 5 10 15 200.0

1.0

2.0

3.0

4.0

Energy (eV)

0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)

n

1

Diffusion in the case of a prism

n(blue) > n(red)

0 5

1.5

2.0


0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

Index of vacuum

0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

No Absorption

Index of refraction : n = 1

« Spring »

Answer to an appliedelectric fied

« An atom »

« Vacuum »

n

k

n

k

Without applied electric field

Energy

Energy


0 5 10 15 200.0

0.5

1.0

1.5

2.0nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0nk« Insulator »n

k

Potentiel électrostatique des noyaux

Electrons liés

Electrostatic Potentialof the nuclei

0 5 10 15 200.0

0.5

1.0

1.5

2.0

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

nk« Semi‐conductor »

k

n

« Insulator »« Semi‐conductor »

(eV)( )Energy (eV)( )Energy

Bounded electrons


0 5 10 15 200.0

0.5

1.0

1.5

2.0nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0nk« Insulator »n

k

Potentiel électrostatique des noyaux

Electrons liés


0 5 10 15 200.0

0.5

1.0

1.5

2.0

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

nk« Semi‐conductor »

k

n

« Insulator »« Semi‐conductor »

(eV)( )Energy (eV)( )Energy

Bounded electrons

« Interior‐sprungMattress »


« Metal »

Collective answer ofan optical excitation


Conduction electron

0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

(eV)

nk

n

k

« Springboard »

Energy


0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)

nk

Index of vacuum

Vacuum

0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)

nk

0 5 10 15 200.0

0.5

1.0

1.5

2.0

Energy (eV)

nk

Insulator Semi‐conductor Metal

Electron more and more free

Absorption band is displaced towards low energies

Directly related to the oscillator strength



Band structure

VB

CB

Eg

1

The different steps to calculate the optical properties:

Imaginary part of the dielectric function: 2

Dipolartransitions

2

Band structure

VB

CB

Eg

1



Real part of the dielectric

function : 1

Kramers-Krönigtransformation3


Dipolartransitions

2

Band structure

VB

CB

Eg

1



4

1 = n2 – k2

2 = 2nk

Determination pfn = f(E) & k = f(E)

Real part of the dielectric

function : 1

Kramers-Krönigtransformation3


Dipolartransitions

2

Band structure

VB

CB

Eg

1



0 5 10

BiVO4-mono atom 0 size 0.70

Γ A Z M L V Γ

E F

Ener

gy (

eV)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

-1.0

-2.0

-3.0

-4.0

-5.0

-6.0

Ener

gy (e

V)

0 2 4 6 8 10 12 140

5

10

Energy (eV)

2

Band structure DOS

Gap 2.3 eV

Band structure calculation1 Determination of 22

Gap 2.3 eV

Optical properties of m-BiVO4 and z-BiVO4 phases

Example of BiVO4


Determination of 13 4 Determination ofn = f(E) & k = f(E)

0

1

2

3

4

0

1

2

3

kn

Energy (eV)0 2 4 6 8 10 12 14

0

5

10

21

-5

0

5

10

Energy (eV)0 2 4 6 8 10 12 14

Example of BiVO4


Journées communications – 15/01/09

5 – OPTICAL PROPERTIES SIMULATION IN WIEN2k

Interband contributions(based on IPA*)

Intraband contributions(using a Drude‐like term)

EF EF

In WIEN2k, two types of contributions to the dielectric function( = 1 + i.2) could be estimated:

h h

Dielectric tensor / Optical conductivity / Refractive index / Reflectivity / Absorption coefficient / Loss function (EELS)

*IPA: Independant Particle Approximation



Interband contributions(based on IPA*)

EF


h h

*IPA: Independant Particle Approximation

Sum over all valence and conduction bands

Im(ij())


5 – INTERPRETATION: Intraband transitions (for metals)

Plasma frequency: (longitudinal oscillations of the electron gas)

Intraband contributions(using a Drude‐like term)

EF



5 – OPTICAL PROPERTIES SIMULATION IN WIEN2kOptical functions:



Symmetry of the dielectric tensor

cubic

monoclinic (, = 90°) orthorhombic

tetragonal, hexagonal

triclinic



Convergence with k‐mesh (expl.: Al)


5 – OPTICS IN WIEN2k

x joint Im ij() (case.joint)

x kram Re ij(), other optical funct.

opticplot

5

6

7


5 – SOME ADDITIONAL DETAILS



The band gap problem Necessity to go beyond DFT

Energy Energy

Electro‐affinity: N+1(N+1) = ‐AIonization energy: N(N) = ‐I

Eg = N+1(N+1) – N(N) = I – A

Energy

Eg = N+1(N) – N(N)

Eg = N+1(2N) – N(2N) + N+1(2N+1) – N+1(2N)

NN‐1

N+1

NN‐1

N+1

NN‐1

N+1

Eg = g + xc

g

with xc : Scissor Operator / GW / hybrid…


Fundamental gap: electronic gap optical gapDeduced from the band structure Dipolar transitions

E

O(2p) Ti(3d)

DO

S

g(1)

2

0 2 4 6 8 10 12 140

2

4

6

8

10

12


If excitonic effects: we should go beyond(TDDFT / BSE)

Fundamental gap: electronic gap optical gapDeduced from the band structure Dipolar transitions

E

O(2p) Ti(3d)

DO

S

g(1)

2

0 2 4 6 8 10 12 140

2

4

6

8

10

12

Energy Energy

NN‐1

N+1

NN‐1

N+1Opticalexcitation

‐30

‐20

‐10

0

10

Energy (eV)

EF

3p Ti

2s O

2p O

3d Ti

Example of the rutile phase

Atomicstructure

Electronicstructure

Dielectricfunction


6 – ILLUSTRATIONS: TiO2 series

‐30

‐20

‐10

0

10

Energy (eV)

EF

3p Ti

2s O

2p O

3d Ti

0 5 10 15 20 25 300

2

4

6

8

10

12

14

Energie (eV)

2EELS experiments



Atomicstructure

Electronicstructure

Dielectricfunction


‐30

‐20

‐10

0

10

Energy (eV)

EF

3p Ti

2s O

2p O

3d Ti

0 5 10 15 20 25 300

2

4

6

8

10

12

14

Energie (eV)

2EELS experimentsDFT simulation



Atomicstructure

Electronicstructure

Dielectricfunction

GGA (PBE)+ gap correction of 0.6 eV




Program flowSCF cycle converged potentialx kgen denser k‐mesh

x lapw1 Kohn‐Sham states for the denser k‐mesh and higher Emax

x lapw2 ‐fermi Fermi distribution

x optic momentum matrix elements (dipolar transitions)

x joint dielectric tensor components: 2

x kram Kramers‐Krönig transformation: 2 1 Optical constants / broadening / scissors operator



TiO2-RUT.inop2000 1 number of k‐points, first k‐point ‐5.0 5.0 Emin, Emax in Ry for matrix elements2 number of choices (columns in *outmat)1 Re xx3 Re zzOFF write unsymmetrized matrix elements to file?

Choices:1......Re <x><x>2......Re <y><y>3......Re <z><z>4......Re <x><y>5......Re <x><z>6......Re <y><z>7......Im <x><y>8......Im <x><z>



TiO2-RUT.injoint1 261 LOWER AND UPPER BANDINDEX0.0000 0.00100 10.0000 EMIN DE EMAX FOR ENERGYGRID IN rydeV output units eV / ryd / cm‐14 SWITCH 2 NUMBER OF COLUMNS0.1 0.1 0.3 BROADENING (FOR DRUDE MODEL ‐ switch 6,7 ‐ONLY)

SWITCH:

0...JOINTDOS FOR EACH BAND COMBINATION 1...JOINTDOS AS SUM OVER ALL BAND COMBINATIONS2...DOS FOR EACH BAND 3...DOS AS SUM OVER ALL BANDS4...Im(EPSILON) 5...Im(EPSILON) for each band combination6...INTRABAND contributions7...INTRABAND contributions including band analysis

TiO2-RUT.inkram0.1 Gamma: broadening of interband spectrum0.6 energy shift (scissors operator)0 add intraband contributions? yes/no: 1/012.60 plasma frequencies (from joint, opt 6)0.20 Gammas for Drude terms



Files generated by:x optic TiO2-RUT.symmat

TiO2-RUT.mommat

x joint TiO2-RUT.joint

x kram TiO2-RUT.epsilon

TiO2-RUT.sigmak

TiO2-RUT.refraction

TiO2-RUT.absorption

TiO2-RUT.eloss

I(є2) V

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

1500 2000 2500 3000 3500 4000 4500

CdTe

CdSe

CdS

ZnTe

ZnSe

ZnS

ZnO

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

52 54 56 58 60 62 64 66 68

Blende structure Wurtzite structure

CdTe

CdSe

CdS

ZnTe

ZnSe

ZnS

ZnO

I(є2)

Refractive index (n)

0 5 10 15 20 250

10

20

30

40

50

2

Energy (eV)

EELS exp.DFT calc.(gap correction)

ZnO Blende ZnS

ZnSe

ZnTe

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.51.61.82.02.22.42.62.83.03.23.43.63.84.04.24.4

Ref

ract

ive

inde

x

Photon Energy (eV)

Ref

ract

ive

inde

x (n

)

Energy (eV)

nCdO 1.95

ZnOZnS

ZnSeZnTe

ZnS

6 –ILLUSTRATIONS: MQ series (M = Zn, Cd & Q = O, S, Se, Te)



7 – Some limitations of DFT simulation of optical properties

‐ Kohn‐Sham eigenstates interpreted as excited states

‐ Independent‐particle approximation (no e‐—h+ interaction)

‐ LDA/GGA are not exact

Use of a scissors operator

Use of Bethe‐Salpeter Equation (BSE) – Time‐dependent DFT

Use of hybrid DFT, effective potentialsUse of DFT+U, LDA+DMFT, GW…

Core level spectroscopy XES, XAS, EELS, XPS

Dipole transitions between core and valence (conduction band or continuum) states

Fermi’s “golden rule”

Time dependent perturbation theory: EM-radiation with energy polarisation and direction of propagation k acts on the momentum p of the electron

tiEtiE eeHtH 0ˆ)(ˆ

The transition probability W from state i to f is then given by Fermi’s “golden rule” :

EEEEwithEfpkeefW ifNixki

fif )()(ˆ 2

W: proportional to the square of the transition matrix element

Number of states E-conservationwith energy E

,

).()(k

txkiekeE

momentum (= dipole) matrix elements:

momentum of photons << momentum of e-; momentum conservation e- cannot change its momentum

...1 xkie rki

dipole quadrupole … approximation

1rkie

1-3% error (even for keV X-rays), but:EELS (electron energy loss spectr.) may violatedipole approximation (selection rules!!)

e-,k

e-,k’

ififif frfefpfefHf ˆˆˆ

selection rules: ℓ±1

XES (X-ray emission spectroscopy)

knock out a core e-

valence e- fills core hole

measure the emitted X-ray

XES gives the ℓ±1 partial DOS of the valence bands of the specific atom (with core state ℓ)

XAS (XANES), EELS (ELNES) core electrons are excited into the

conduction band

Each core shell introduces an absorption edge, (they are indexed by the principal number of a core level)

core levels

2p1/2

2p3/2SOC

2s1sK

L1

L3L2

K-1s, L1-2s, L2-2p1/2, L3-p3/2

XAS: XES gives the ℓ±1 partial DOS of the conduction bands of the specific atom (with core state ℓ)

h

Difference between EELS and XAS

XAS vs. EELS: theory

transition described by Fermis “golden rule” between initial (core) and final (conduction-band) state and the e- or photon

double differential cross section:E - conservation

single diff. cross section

momentum transfer q polarization vector e

dipole approximation

core-valence spectroscopies give informationon the local DOS (because of <core|r|val>) of angular momentum character ℓ ± 1

(TELNES3 can also handle non-dipole transitions + relativistic corrections)

“Final state rule”:

“Final state” determines the spectrum:•Absorption spectroscopy:

Final state has a “hole” in core state, but additional e- in conduction band. Core-hole has large effect on the spectrum

• electron – hole interaction, “excitonic effects”

• Emission spectroscopy:Final state has filled core, but valence hole.This is usually well screened, thus one “sees” the groundstate.

.

“Final state rule” + core hole:

2x2x2 supercell calculation with core hole in one of the Mg atoms(add e- to valence or “background”).

This allows the conduction state torelax (adjust to the larger effectivenuclear charge), but also to have static screening from the environment.

core hole, no supercell:

Z+1 (AlO)

groundstate

MgO

exp

Core hole calculations in WIEN2k

label core-hole atom „Mg1“

edit case.inc and case.in2 (inm)

(needs case.inxs)

EELS in WIEN2k

supercell calculations as for XAS TELNES3 task in w2web

When IPA fails: Electron – hole interactions

when the e- is not ionized, but stays in the solid:

excitonic effects Frenkel (localized) and Wannier (delocalized) excitons

independent particle approx.

importance of excitons (BSE):

L2,3 spectra: failure of the single particle approach

In particular early 3d TM-compounds show a non-standard L2/L3 branching ratio (1:2) sometimes a completely different lineshape (TiO2) „wrong“ SOC or CF splittings

rutile TiO2 CaF2

exp.exp.

soc

core-hole calc.

ground state calc.

L3 L2

CF

fully relativistic electron-hole interaction (BSE)

Bethe-Salpeter-equation: L(12;1’2’) solving a 2-particle (e- - h) equation of

large dimension ( Nv Nc Nk ~ 100000)

)'()'()',()()('

)'()()',()'()('

)(2

''*

''*33

'''

''*

''*33

'''

''',,

rrrrvrrrdrdH

rrrrWrrrdrdH

EEHHHHH

kckvckvkx

kcvckv

kckvckvkdir

kcvckv

kkvvcckckvdiag

xdirdiageh

eigenvalue difference between hole (v) and electron(c) state

attractive screened static Coulomb interaction W; W~-1

e-h exchange with bare Coulomb potential v

Ca-L23 edge in CaF2

“ground-state” DOS

“core-hole” calc.(ratio 2:1)

BSE for L2 and L3 separately

experiment

BSE with p1/2 and p3/2 together

BSE without direct term Hehdir

BSE without exchange term Hehx

L2,3 edge for Ca in CaF2

Decomposition of є2 into the excitation from p1/2 and p3/2 statescross terms suppress the L3 branch and enhance L2

L3 L2

XPS, core-level shifts

Ionizationpotential of core-e-, IP= Etot(N) – Etot(N-1) gives information on charge state of the atom

core-eigenvalues i (with respect to EF) are NOT a good approximation: i=dE/dn Janak’s theorem

Slater’s “transition state”: core-eigenvalues i for half occupancy

-SCF-calculation with and without core-hole: Etot(N) – Etot(N-1) always use supercells to reduce hole-hole interaction

C,N 1s exp.(eV) i -SCF

TiC 281.5 264.7 281.9

Ti4C4 281.5 263.3 281.1

TiN 397.0 377.5 397.1

E

En

N-1 N

secant ~ tangent at N-½

Band gap problem, independent particle approx.

better local potentials: TB-mBJmany-body perturbation theory: GW approach

shift of conduction bands: scissors operator

Electron-affinityIonization energy

Band gap

GW (MBPT on one page)

Self-energy

Green’s function G:

screened coulomb interaction W:

GW: screened HF approximation + calculation of “quasiparticle energies”:

describes propagation of e-,h+

from r1r2 with time ()

GW and BSE are available upon request (see „unsupported software page“ at wien2k.at

Both calculations are EXTREMELY expensive Hardware: at least 128 cores with infiniband

Optical properties, Core-level spectroscopy, GW and BSE

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Transcript of Optical properties, Core-level spectroscopy, GW and BSE