X-ray spectroscopies on correlated systems ------------ Hubbard model for H 2 molecule / MIT in Ti 2 O 3 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
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 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
Standard model: one-electron approximation 1. Free electron theory for metals (Kittel chapter 6). “density of states” 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.
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 !!
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
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 !!!
Photoelectric effect: kinetic energy monochromatic to be analyzed light Conservation of energy: • near sample : E kinetic = h ν – E binding - Φ sample • near analyzer: E kinetic = h ν – E binding - Φ analyzer
One-particle approximation • Photoelectron Spectroscopy -- “occupied” density of states • Inverse Photoelectron Spectroscopy -- “unoccupied” density of states
Clean polycrystalline silver h ν = 1486.6 eV Ag: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 1 3d 5/2 core levels 3d 3/2 valence band 3p 3/2 3p 1/2 3s 4d 4s 4p Fermi level
Clean polycrystalline silver h ν = 1486.6 eV Ag valence band Ag 4d Fermi cut-off Ag is a metal Ag 5sp
Chemical composition of chromium oxide grown on MgO using NO 2 Cr 3p 3/2 Cr 3p 1/2 O 1s Cr 2s N 1s Cr 3p Cr 3s
Chemical Environment
Can we understand the spectral lineshape ? Binding Energy (eV) Why extra peaks in Cu 2p core level of CuO ?
Can we understand the spectral lineshape ? exp LDA exp LDA exp LDA Binding Energy (eV) Why extra high energy peaks in VB of CuO ?
True ! These are incorrect statements: “ground state d-bands“ is an invalid concept !! This is an incorrect statement: “initial state d-bands“ is an invalid concept !!
This is an incorrect statement: “ground state d-bands“ do not exists and can therefore not be measured True ! 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"
H 2 model (a very simple case study) one-electron approximation vs. Hubbard model
H 2 molecule model one-electron approximation energy level diagram spectrum inverse photoemission photoemission a b 2t 2t E F triplet singlet E t – E s = 2t 2t 2t too large !!
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H 2 molecule model Hubbard model total energy level diagram N=2 a 2 b 0 , a 0 b 2 U a 1 b 1 a 1 b 1 E t –E s ~ 4t 2 /U GS (triplet) (singlet)
H 2 molecule model Hubbard model total energy level diagram N=1 (PES) N=2 N=3 (IPES) a 1 b 0 , a 0 b 1 a 2 b 1 , a 1 b 2 2t 2t a 2 b 0 , a 0 b 2 U a 1 b 1 GS (singlet)
H 2 molecule model : Hubbard model S M M S anti-bonding bonding bonding anti-bonding E F U 2 + 16t 2 - 2t E gap =
PES/IPES spectral weights
- δ =
H 2 molecule model : Hubbard model
Ti 2 O 3 electronic structure and dimer formation
Metal-Insulator-Transition in Ti 2 O 3 Ti 2 O 3 gradual transition ~10 1 change in ρ ~10 1 500 K 300 K
c -axis dimer Ansatz for Ti 2 O 3 • Ti 3+ : 3d 1 , S=1/2 • Ti 3+ -Ti 3+ pairs : a 1g molecular singlet formation effectively S=0
Determining orbital occupations: soft-x-ray absorption spectroscopy • use of core levels → local transitions → element and site specific E Fermi • involves most relevant orbitals: Ti 3d O 2p 2p-3d (TM), 3d-4f (RE), 1s-2p (O,N,C) • dipole allowed → very strong intensities • dipole selection rules + multiplet structure h ν ≈ 460 eV h ν ≈ 530 eV give extreme sensitivity to symmetry of initial state: charge , spin and orbital Ti 2p 3/2 2p 1/2 theory: O 1s • Cluster calculations with full atomic- multiplet theory • LDA, contrained LDA+U calculations Spectrum ( h ν) = Σ f 〈 i e.r f 〉 ² δ ( h ν - E f + E i ) provide input parameters i 〉 = many-body initial state, f 〉 = many-body final state Near ground state properties and spectra e.r = dipole transition are treated on equal many-body footing
Theoretical sensitivity of XAS to orbital occupation in Ti 2 O 3 E ⊥ C E ⊥ C π a 1g a 1g a 1g e E II C E II C g diff. diff. Intensity (arb. units) Intensity (arb. units) 455 460 465 470 455 460 465 470 Energy (eV) Energy (eV) Different orbital occupation has different spectral features and polarization dependence !
Experimental orbital occupation in Ti 2 O 3 in the insulating phase E ⊥ C E ⊥ C Experiment a 1g a 1g E II C T= 300 K E II C diff. diff. Intensity (arb. units) 455 460 465 470 455 460 465 470 Energy (eV) Energy (eV) Insulating state: Ti 3+ -Ti 3+ c-axis dimers are electronically formed !!
Experimental orbital occupation in Ti 2 O 3 in the insulating phase E ⊥ C E ⊥ C Experiment a 1g a 1g E II C T= 300 K E II C diff. diff. Intensity (arb. units) 455 460 465 470 455 460 465 470 Energy (eV) Energy (eV) Insulating state: Ti 3+ -Ti 3+ c-axis dimers are electronically formed !! Not so for V 2 O 3 !!!
c -axis dimer Ansatz for Ti 2 O 3 • Ti 3+ : 3d 1 , S=1/2 • Ti 3+ -Ti 3+ pairs : a 1g molecular singlet formation effectively S=0
valence band photoemission on Ti 2 O 3 single crystals U/t = 0 room temperature : insulating phase U/t = 1 2t U/t ∼ 3 U/t = 5 anti- bonding bonding U/t = 10 O 2p Ti 3d U/t = 100 Two-peak structure like in a H 2 molecule model (relative weights according to quantum mechanical interference effect)
Ti 2p core level XPS: experiment vs. multiplet theory single cluster (Ti-O6) c-axis dimer single-site
Ti 2p core level XPS: experiment vs. multiplet theory
What happens across the Metal Insulator Transition ? Ti 2 O 3 gradual transition ~10 1 change in ρ 300 K 500 K
Orbital occupation in Ti 2 O 3 from XAS: across MIT Experiment E ⊥ C Theory E ⊥ C E II C E II C 48.7% a 1g a 1g Intensity (arb. units) 575 K Intensity (arb. units) 71.5% a 1g a 1g 500 K 77.8% a 1g a 1g 458 K a 1g a 1g 300 K 455 460 465 470 455 460 465 470 Energy (eV) Energy (eV)
Orbital occupation in Ti 2 O 3 from XAS: across MIT “dimer” “isotropic” 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 Ti 2 O 3 )
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