Hao Tjeng

Max-Planck-Institute Chemical Physics of Solids

Dresden

• Roger Chang, Zhiwei Hu, Stefano Agrestini, Jonas Weinen, Maurits Haverkort, Li Zhao, Alexander Komarek – MPI Dresden • Yen-Fa Liao, Ku-Ding Tsuei, Hong-Ji Lin, Chien-Te Chen – NSRRC, Taiwan • Thomas Koethe, Holger Roth, Thomas Lorenz – Univ. Cologne • Giancarlo Pannaccione – Elettra, Trieste • Arata Tanaka – Univ. Hiroshima, Japan

X-ray spectroscopies on correlated systems ------------

Hubbard model for H2 molecule / MIT in Ti2O3

Electronic structure = ground state + electronic excitations: • excitations, density of states, spectral weight • charge neutral: temperature, thermodynamics • charge neutral: optical spectroscopy, x-ray absorption spectroscopy • electron removal: photoelectron spectroscopy • electron addition: inverse photoelectron spectroscopy

Importance of electronic structure: • crystal structure, atomic structure • physical properties determined by electrons • nucleus is very small, spherical, gives only mass and positive charge • electrons determines crystal structure and all physical properties

Standard model: one-electron approximation

The idea is to calculate first the 1-electron states, and then fill it up from the lowest states on, so that all N electrons are accommodated. The highest state occupied then defines the Fermi level.

1. Free electron theory for metals (Kittel chapter 6).

“density of states”

Standard model: one-electron approximation

• The Fermi function gives the probability that a particular one-electron state at energy E is occupied by an electron. • The Fermi level gives distinction between occupied and unoccupied density of states.

Standard model: one-electron approximation

2. Crystals, energy bands (Kittel ch. 7, Ashcroft+Mermin ch. 9, 10, 11 )

Again: The idea is to calculate first the 1-electron states, and then fill it up form the lowest states on, so that all N electrons are accommodated. The highest state occupied then defines the Fermi level.

In general: any band structure calculation follows this recipe !!

All semiconductor physics is based on 1-electron theory !!

3. Wavefunctions: from N-electron to N 1-electron (Ashcroft+Mermin chapter 17)

For the ground state – ground state energy and charge density • The N-electron wave functions can be expressed as a product of N 1-electron wave functions • Each 1-electron wave function needs to be ‘optimized’ • The Slater determinant is to ensure Pauli principle

Standard model: one-electron approximation --- why so popular ??

3. Wavefunctions: from N-electron to N 1-electron (Ashcroft+Mermin chapter 17)

For the ground state – ground state energy and charge density • The N-electron wave functions can be expressed as a product of N 1-electron wave functions • Each 1-electron wave function needs to be ‘optimized’ • The Slater determinant is to ensure Pauli principle

!!! It should be forbidden to look inside the Slater determinant !!!

Standard model: one-electron approximation --- why so popular ??

Conservation of energy: • near sample : Ekinetic = hν – Ebinding - Φsample • near analyzer: Ekinetic = hν – Ebinding - Φanalyzer

monochromatic light

kinetic energy to be analyzed

Photoelectric effect:

One-particle approximation • Photoelectron Spectroscopy -- “occupied” density of states • Inverse Photoelectron Spectroscopy -- “unoccupied” density of states

Clean polycrystalline silver

Ag: 1s22s22p63s23p63d104s24p64d105s1

valence band

hν = 1486.6 eV

Fermi level

3p1/2 3p3/2

3s

3d3/2

3d5/2

4s 4p 4d

core levels

Clean polycrystalline silver

Ag valence band

Fermi cut-off

Ag 4d

Ag 5sp Ag is a metal

hν = 1486.6 eV

Chemical composition of chromium oxide grown on MgO using NO2

Cr 2s

Cr 3p1/2

Cr 3p3/2

Cr 3s Cr 3p

O 1s

N 1s

Chemical Environment

Why extra peaks in Cu 2p core level of CuO ?

Binding Energy (eV)

Can we understand the spectral lineshape ?

Why extra high energy peaks in VB of CuO ?

Can we understand the spectral lineshape ?

Binding Energy (eV)

exp LDA

exp LDA

exp LDA

These are incorrect statements: “ground state d-bands“ is an invalid concept !!

True !

This is an incorrect statement: “initial state d-bands“ is an invalid concept !!

True !

This is an incorrect statement: “ground state d-bands“

do not exists and can therefore not be measured

This is an incorrect statement: “ground state d-bands“

do not exists and can therefore not be measured

Ashcroft and Mermin: Solid State Physics, page 309

"unoccupied"

"occupied"

H2 model

(a very simple case study)

one-electron approximation vs. Hubbard model

a b

energy level diagram

H2 molecule model one-electron approximation

2t

spectrum

2t

EF

photoemission inverse

photoemission

2t 2t triplet singlet Et – Es = 2t

too large !!

A{ }

A{ }

A{ }

H2 molecule model Hubbard model

total energy level diagram

a1b1

a2b0, a0b2

U

GS

N=2

(singlet)

a1b1

(triplet) Et–Es ~ 4t2/U

H2 molecule model Hubbard model

total energy level diagram

a1b0, a0b1 2t

N=1 (PES)

a2b1, a1b2 2t

N=3 (IPES)

a1b1

a2b0, a0b2

U

GS

N=2

(singlet)

H2 molecule model : Hubbard model

EF

M S

bonding anti-bonding bonding anti-bonding

S M

Egap = U2 + 16t2 - 2t

PES/IPES spectral weights

δ = -

H2 molecule model : Hubbard model

Ti2O3 electronic structure and dimer formation

Ti2O3

~101

500 K 300 K

gradual transition ~101 change in ρ

Metal-Insulator-Transition in Ti2O3

c-axis dimer Ansatz for Ti2O3

• Ti3+ : 3d1, S=1/2 • Ti3+-Ti3+ pairs : a1g molecular singlet formation effectively S=0

Determining orbital occupations: soft-x-ray absorption spectroscopy

EFermi

Ti 2p3/2 2p1/2

O 1s

O 2p Ti 3d

hν ≈ 460 eV hν ≈ 530 eV

Spectrum (hν) = Σf⟨ie.rf⟩² δ(hν - Ef + Ei) i⟩ = many-body initial state, f⟩ = many-body final state e.r = dipole transition

• use of core levels → local transitions → element and site specific • involves most relevant orbitals: 2p-3d (TM), 3d-4f (RE), 1s-2p (O,N,C) • dipole allowed → very strong intensities • dipole selection rules + multiplet structure give extreme sensitivity to symmetry of initial state: charge, spin and orbital

theory: • Cluster calculations with full atomic- multiplet theory • LDA, contrained LDA+U calculations provide input parameters

Near ground state properties and spectra are treated on equal many-body footing

455 460 465 470

a1ga1g

Intensity (arb. units)

Energy (eV)

Inte

nsity

(arb

. uni

ts)

Energy (eV)455 460 465 470

a1geπg

Different orbital occupation has different spectral features and polarization dependence !

Theoretical sensitivity of XAS to orbital occupation in Ti2O3

E ⊥ C E II C diff.

E ⊥ C E II C diff.

455 460 465 470

a1ga1g

Inte

nsity

(arb

. uni

ts)

Energy (eV)

E ⊥ C E II C diff.

Energy (eV)

455 460 465 470

E ⊥ C E II C diff.

ExperimentT= 300 K

Insulating state: Ti3+-Ti3+ c-axis dimers are electronically formed !!

Experimental orbital occupation in Ti2O3 in the insulating phase

455 460 465 470

a1ga1g

Inte

nsity

(arb

. uni

ts)

Energy (eV)

E ⊥ C E II C diff.

Energy (eV)

455 460 465 470

E ⊥ C E II C diff.

ExperimentT= 300 K

Insulating state: Ti3+-Ti3+ c-axis dimers are electronically formed !! Not so for V2O3 !!!

Experimental orbital occupation in Ti2O3 in the insulating phase

c-axis dimer Ansatz for Ti2O3

• Ti3+ : 3d1, S=1/2 • Ti3+-Ti3+ pairs : a1g molecular singlet formation effectively S=0

valence band photoemission on Ti2O3 single crystals

bonding anti-

bonding

U/t = 0

U/t = 1

U/t = 5

U/t = 10

U/t = 100

Two-peak structure like in a H2 molecule model (relative weights according to quantum mechanical interference effect)

O 2p Ti 3d

2t U/t ∼ 3

room temperature : insulating phase

single cluster (Ti-O6)

Ti 2p core level XPS: experiment vs. multiplet theory

c-axis dimer

single-site

Ti 2p core level XPS: experiment vs. multiplet theory

What happens across the Metal Insulator Transition ?

Ti2O3

500 K 300 K

gradual transition ~101 change in ρ

455 460 465 470

E ⊥ C E II C

575 K

500 K

458 K

300 K

Experiment

In

tens

ity (a

rb. u

nits

)

Energy (eV)455 460 465 470

Intensity (arb. units)

48.7% a1ga1g

71.5% a1ga1g

77.8% a1ga1g

a1ga1g

E ⊥ C E II C

Theory

Energy (eV)

Orbital occupation in Ti2O3 from XAS: across MIT

“dimer” “isotropic”

Orbital occupation in Ti2O3 from XAS: across MIT

Break-up of “dimers“

MIT = going from a collection of “dimers“ into a 3-dimensional solid

“Making hydrogen metallic“ (but there are no orbital degrees of freedom in hydrogen,

while orbital degrees of freedom are essential for the MIT in Ti2O3)

300K 450K 575K 320K

Inte

nsity

(Arb

. uni

ts)

Ti2O3 VB

HAXPES hv = 6.5 keV

Ti2O3 : single crystal, valence band

16 14 12 10 8 6 4 2 0 -2 -4

Binding Energy (eV)

hν 6.5keV

O 2p Ti 3d

Inte

nsity

(Arb

. uni

ts)

300K 450K 575K 320K Au ref.

Ti2O3 VB2,0 1,5 1,0 0,5 0,0 -0,5

Binding Energy (eV)

Ti2O3 : single crystal, Ti 3d

HAXPES hv = 6.5 keV

Gradual MIT – bad metal in metallic phase

The terms occupied and unoccupied density of states are very differently defined in many body language ! At T=0 only the ground state is occupied, and all the other states are unoccupied. The ground state is only ONE state, and has no band structure nor density of states nor band gaps nor exchange splitting nor …… splitting. The many body electron system obeys the Fermi-Dirac statistics, but the Fermi-Dirac distribution function applies only for 1-electron systems – The Fermi function is a 1-electron theory concept !

Important Remarks:

Photoemission and Inverse Photoemission cannot be described as measuring the “occupied and unoccupied density of states”. Optical spectroscopy cannot be interpreted as measuring the “joint density of states”.

Important Remarks:

extra peaks in Cu 2p core level of CuO due to Coulomb interaction between the Cu 2p core hole and the Cu 3d valence hole (d9)

Binding Energy (eV)

Can we understand the spectral lineshape ?

Concluding remarks : Photoelectron Spectroscopy: ideal to study electron correlations

H2 Hubbard model

One-particle approximation

• photoemission: --- “occupied” density of states • inverse photoemission: --- “unoccupied” density of states

Many-body framework

photoemission: “valence band”

inv. photoemission: “conduction band”

ω

ρ k(ω) = 1/π * |Im Gk (ω)|