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Introduction to crystal field multiplet calculations Introduction - - PowerPoint PPT Presentation

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Introduction to crystal field multiplet calculations Introduction Introduction Multiplets (atomic physics) Multiplets (atomic physics) Multiplets (crystal field) Multiplets (crystal field) 1 2p XAS NiO 2p - XAS Experiment is sharper than

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Introduction to crystal field multiplet calculations

Multiplets (atomic physics) Multiplets (atomic physics) Multiplets (crystal field) Multiplets (crystal field) Introduction Introduction

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NiO 2p - XAS L edge

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L edge

2

L3 edge jump continuum L2 edge jump continuum Intensity (arb. units) 840 845 850 855 860 865 870 875 880 885 Photon energy (eV)

2p XAS

2p

Experiment is sharper than LDA empty density of states Experiment is sharper than LDA empty density of states Branching ratio (L2, L3 intensity ratio is not 3:2) Branching ratio (L2, L3 intensity ratio is not 3:2) Experiment shows more structure (multiplets) than DOS Experiment shows more structure (multiplets) than DOS

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  • Excitons and multiplets dominate the spectral line-

shape.

  • It is not sufficient to assume independent electrons

interacting with an average potential

  • One has to consider the interaction between each

pair of electrons explicitly.

2p XAS (core level spectroscopy)

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Charge Transfer Multiplet program Quanty, CTM4XAS, Tanaka code (XTLS), etc Used for the analysis of XAS, EELS, PES, IPES, Auger, RIXS, NIXS, etc ATOMIC PHYSICS ⇓ CRYSTAL FIELD THEORY ⇓ CHARGE TRANSFER

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Quanty - XAS

PRB 85, 165113 (2012) PRB 93, 165107 (2016)

M45 edges rare-earth M45 edges rare-earth L23 edges transition metals L23 edges transition metals

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Quanty – RIXS

Orbital and dispersing magnetic transitions in TiOCl Orbital and dispersing magnetic transitions in TiOCl

PRL 107, 107402 (2011).

Linear Dichroism in Sr2IrO4 Linear Dichroism in Sr2IrO4

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Quanty – non-resonant IXS

Angular dependence of d-d excitations in NiO Angular dependence of d-d excitations in NiO

EPL 96, 37007 (2011)

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Charge Transfer Multiplet program input

  • Initial State configuration (3d8 for Ni2+)
  • Final state configuration (2p53d9 for Ni2+ 2p XAS)
  • Transition operator (dipole for XAS)
  • Experimental geometry (polarization of the light, sample
  • rientation, magnetic field, temperature, etc.)
  • Hi Hamiltonian of the initial state
  • Hf = Hi + Hcd + HSOC

Hamiltonian of the final state

core

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Charge Transfer Multiplet program What information we can get ?

  • Ground state symmetry, important quantum numbers,

magnetic moment, etc.

  • Orbital occupation and orbital level splitting
  • Many body energy level diagram
  • Physical properties (magnetic anisotropy, susceptibility,

magnetization, g factors, etc.)

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Hamiltonian

  • + ξ ∙ +
  • + +

∙ + 2

=

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Hamiltonian (atomic physics)

  • + ξ ∙ +
  • + +

∙ + 2

=

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Coulomb interaction

Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. with with

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Coulomb interaction

Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. with with

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Coulomb interaction – Slater Integrals

Expansion on renormalized Spherical Harmonics Expansion on renormalized Spherical Harmonics Integral to calculate Integral to calculate

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Coulomb interaction – Slater Integrals

Radial part: Slater integrals Radial part: Slater integrals Angular part: Analytical solution Angular part: Analytical solution

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Coulomb interaction – Slater Integrals

d electrons d electrons f - electrons f - electrons

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Coulomb interaction – Slater Integrals

Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – exchange term Core (p) valence (d) interaction – exchange term

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Coulomb interaction – Slater Integrals

Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – exchange term Core (p) valence (d) interaction – exchange term

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Initial state Hamiltonian (atomic multiplet theory)

Valence Spin-orbit coupling Valence Spin-orbit coupling

ξ ∙

  • Electron-electron interaction of

valence states Electron-electron interaction of valence states

∑ ∑

+ =

+ + k k k k k k J S r e J S

G g F f L L

1 2 1 2

| |

12 2

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Valence Spin-orbit coupling Valence Spin-orbit coupling

ξ ∙

  • Electron-electron interaction of

valence states Electron-electron interaction of valence states

Final state Hamiltonian (atomic multiplet theory)

ξ ∙

  • Core-valence electron interaction

Core-valence electron interaction Core Spin-orbit coupling Core Spin-orbit coupling

∑ ∑

+ =

+ + k k k k k k J S r e J S

G g F f L L

1 2 1 2

| |

12 2

∑ ∑

+ =

+ + k k k k k k J S r e J S

G g F f L L

1 2 1 2

| |

12 2

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Valence Spin-orbit coupling Valence Spin-orbit coupling

ξ ∙

  • Electron-electron interaction of

valence states Electron-electron interaction of valence states

Final state Hamiltonian (atomic multiplet theory)

ξ ∙

  • Core-valence electron interaction

Core-valence electron interaction Core Spin-orbit coupling Core Spin-orbit coupling

∑ ∑

+ =

+ + k k k k k k J S r e J S

G g F f L L

1 2 1 2

| |

12 2

∑ ∑

+ =

+ + k k k k k k J S r e J S

G g F f L L

1 2 1 2

| |

12 2

Spin-orbit coupling couples L and S quantum numbers Only the total moment J is a good quantum number

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Atomic multiplet theory

Ca 2p XAS Ca 2p XAS

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Term symbols

2S+1L

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Term symbols – single electron

Orbital occupation Orbital occupation

s1 S=1/2, L=0 →

2S

Angular momentum Angular momentum Term symbol Term symbol

p1 S=1/2, L=1 →

2P

d1 S=1/2, L=2 →

2D

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Term symbols – two electrons

Orbital occupation Orbital occupation

1s12s1

1S, 3S

Angular momentum Angular momentum Term symbol Term symbol

L1s=0, L2s=0 → Ltot=0 S1s=1/2, S2s=1/2 → Stot=0,1

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Term symbols – two electrons

Orbital occupation Orbital occupation

1P, 1D, 1F

Angular momentum Angular momentum Term symbol Term symbol

L2p=1, L3d=2 → Ltot=1, 2 or 3 S2p=1/2, S3d=1/2 → Stot=0,1

3P, 3D, 3F

(2p53d1) 2p13d1

2S+1L

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Term symbols – two electrons

Orbital occupation Orbital occupation Angular momentum Angular momentum Term symbol Term symbol

S3d=1/2, L3d=2 → J3d=3/2,5/2 S2p=1/2, L2p=1 → J2p=1/2,3/2

(2p53d1) 2p13d1

For each term symbol JMAX= S1 + L1 JMIN= |S1 – L1| Steps of 1 between JMIN and JMAX

1P1, 3P0, 3P1, 3P2 1D2, 3D1, 3D2, 3D3 1F3, 3F2, 3F3, 3F4

2S+1LJ

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Atomic multiplet theory

3 peaks in the spectrum. Why?

Ca 2p XAS Ca 2p XAS

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Atomic multiplet theory

Ca 2p XAS Ca 2p XAS

3 peaks in the spectrum. Why? 3d0→2p53d1 Dipole transition Without spin-orbit coupling Initial state symmetry :

1S

Final state symmetry :

1P, 1D, 1F

Selection rules for dipole transition: ΔL=+1 or -1 ΔS=0 Allowed transition <1S| ∆S=0;∆L=+1 | 1P> ≠0 1 peak in the spectrum

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Atomic multiplet theory

Ca 2p XAS Ca 2p XAS

3 peaks in the spectrum. Why? 3d0→2p53d1 Dipole transition With spin-orbit coupling Initial state symmetry :

1S0

Final state symmetry :

1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2, 3D3 1F3, 3F2, 3F3, 3F4

Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 3 peaks in the spectrum

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Atomic multiplet theory

Ca 2p XAS Ca 2p XAS

3d0→2p53d1 Dipole transition With spin-orbit coupling Initial state symmetry :

1S0

Final state symmetry :

1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2, 3D3 1F3, 3F2, 3F3, 3F4

Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 3 peaks in the spectrum

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Atomic multiplet theory

4f0→3d94f1 Dipole transition With spin-orbit coupling Initial state symmetry :

1S0

Final state symmetry :

1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2, 3D3 , 1F3, 3F2, 3F3, 3F4, 1G4, 3G3, 3G4, 3G5, 1H5, 3H4, 3H5, 3H6

Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 3 peaks in the spectrum

La 3d XAS La 3d XAS

Calculated Experimental

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Atomic multiplet theory

4f0→3d94f1 Dipole transition With spin-orbit coupling Initial state symmetry :

1S0

Final state symmetry :

1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2, 3D3 , 1F3, 3F2, 3F3, 3F4, 1G4, 3G3, 3G4, 3G5, 1H5, 3H4, 3H5, 3H6

Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 3 peaks in the spectrum

La 3d XAS La 3d XAS

Calculated Experimental

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Atomic multiplet theory

Ti4+ 2p XAS Ti4+ 2p XAS

3d0→2p53d1 Dipole transition With spin-orbit coupling (SOC) Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 Atomic multiplet theory predicts 3 peaks in the spectrum Calculated Calculated Fk, Gk + SOC

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Atomic multiplet theory

Ti4+ 2p XAS Ti4+ 2p XAS

3d0→2p53d1 Dipole transition With spin-orbit coupling Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions <1S0|∆J=+1| 1P1, 3P1 , 3D1> ≠0 Atomic multiplet theory predicts 3 peaks in the spectrum

Experimentally 6 peaks are observed !!!!

Calculated Fk, Gk + SOC

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Crystal Field theory

d orbitals d orbitals

t2g states eg states

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Crystal Field theory

Oh symmetry (octahedral crystal field) Oh symmetry (octahedral crystal field)

  • eg: x2-y2, z2
  • 10Dq

d orbitals t2g: yz, xz, xy

Spherical Spherical

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Crystal field multiplet theory

Ti4+ 2p XAS Ti4+ 2p XAS

3d0→2p53d1 Dipole transition We need to go beyond the atomic multiplet theory and include the crystal field (CF) in the Hamiltonian

  • + ξ ∙ +
  • 38
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Crystal field multiplet theory

Ti4+ 2p XAS Ti4+ 2p XAS

3d0→2p53d1 Dipole transition We need to go beyond the atomic multiplet theory and include the crystal field (CF) in the Hamiltonian

  • + ξ ∙ +
  • Calculated

Fk, Gk + SOC + CF eg t2g eg t2g

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Crystal field multiplet theory

Co3+ 2p XAS Co3+ 2p XAS

3d6→2p53d7 Dipole transition

S=0 (low spin) S=2 (high spin)

eg t2g

775 780 785 790 795 800 805

Energy (eV) Sr2Co0.5Ir0.5O4 Co 2p1/2 Co3+ S=2 Co 2p3/2

775 780 785 790 795 800 805

Co 2p1/2 Co3+ S=0 Energy (eV) NdCaCoO4 Co 2p3/2

  • 10Dq

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Crystal Field theory

Oh symmetry (octahedral crystal field) Oh symmetry (octahedral crystal field)

  • eg: x2-y2, z2
  • 10Dq

d orbitals t2g: yz, xz, xy

Spherical Spherical D4h symmetry (tetragonal crystal field) D4h symmetry (tetragonal crystal field)

∆eg>0 ∆t2g>0

b1g: x2-y2 a1g: z2 b2g: xy eg: xz, yz

compressed ∆eg<0 ∆t2g<0

b1g: x2-y2 a1g: z2 b2g: xy eg: xz, yz

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Crystal Field theory

Ni 2p XAS of NiO film Ni 2p XAS of NiO film 3d8 in D4h symmetry (tetragonal crystal field) 3d8 in D4h symmetry (tetragonal crystal field)

∆eg=0.48 eV ∆t2g=0.36 eV

b1g: x2-y2 a1g: z2 b2g: xy eg: xz, yz

  • M. Haverkort et al. PRB 69, 020408R (2004)

Ds=0.12 eV, Dt=0 eV

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Crystal Field theory

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