norm (cps)

Correlated Electrons

In Complex Transition Metal Oxides

Prof. Thomas Brückel IFF - Institute for Scattering Methods

& RWTH Aachen - Experimental Physics IV c

Neutron Laboratory Course 2007 Correlated Electron Systems

Strongly correlated electrons: movement of one electron depends on positions of all other electrons due to long ranged Coulomb repulsion Novel Phenomena and functionalities:

• high temperature superconductivity (1986: Bednorz & Müller)

• colossal magneto resistence CMR

• magnetocaloric effect

• multiferroic effect

• metal-insulator transition

• negative thermal expansion

• ???

for you to discover

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• summary

Electronic Structure of Solids

• adiabatic approximation(Born-Oppenheimer)

separates lattice and electronic degrees of freedom

• Fermi gas: free electron model: single electron moves in 3d potential well with infinitely high walls (crystal surfaces)

• Fermi liquid: electron-electron interaction accounted for by quasiparticles

“dressed electrons” with charge e, spin ½, but effective mass m*

• band structure: takes into account periodic potential of atomic cores at rest;

e^-moves in average potential from atomic cores and other e^-

pot. energy free electrons:

potential well atomic core pot.

single particle wave function

• electronic correlations: strong Coulomb interaction! Model (LDA+U; DMFT,…) ?

Band Structure of Solids

tight binding model:

delocalization

nonmagnetic magnetic

itinerant localized Width of band structures W for trans. & RE metals:

Width of electronic bands:

Band Structures and Conductivity

semi conductor conduction

band

valence band

core level Fermi energy

E

metal insulator

… but where are the electronic correlations?

Fermi- Dirac distrib.

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• summary

Breakdown of Band Theory

Typical example: transition metal oxides e. g. CoO CoO: rock salt structure →1 Co & 1 O per unit cell electron configuration: Co: [Ar] 3d⁷4s²

O: [He] 2s²2p⁴

⇒ total number of electrons per unit cell: 9 + 6 = 15

uneven number of electrons →at least one partially filled band (spin up and down!)

→CoO≡metal !

in reality: CoO≡ insulator (ρ ≈10⁸Ωcm @ RT; compare: Fe → ρ ≈10^-7Ωcm) with activation energies ≈0.6 eV≈7000 K !

LDA: doubtful that insulating character can be reproduced

Mott Transition

Tight-binding picture of band structure of Na: [Ne] 3s¹= 1s²2s²2p⁶3s¹

ok but should hold for a →∞ 3s-band is half filled ⇒ Na ≡metal

according to Heisenberg Δ ⋅Δ ≥p x / 2 we gain in kinetic energy if electrons are delocalized

conductivity is connected with charge fluctuations:

⇒ charge transfer costs energy U (1 … 10 eV)

→Mott transitionfrom metal to insulator for a critical value of a

Na⁰ Na⁰ e^-

Na⁺ Na^-

ε3s ε3s O 2ε3s + U3s

single particle energy for 3s electron

intraatomic Coulomb repulsion hopping t

Hubbard-Model: "Lattice Fermion Model"

single band Hubbard Hamiltonian:

(in second quantization)

+:

σ

cj σ: nj

creates electron in tight binding (Wannier)-stateΦ(r−Rj)σ occupation operatorc⁺j_σcj_σ

U : Coulomb repulsion in one orbital: ⁼

∫ ∫

2 2 2 1 2 2

1 4

) ( ) (

r r

R r R r e dr dr

U ^{j j}

πε

•Simplest way to incorporate correlations due to Coulomb-interaction:

only the strongest contribution (on-site interaction ≈ 20 eV) is taken into account.

•Rich physics: FM / AF metals & insulators, charge and spin density waves, …

•Realistic Hamiltonian should contain many intersite terms (Coulomb-interaction is long ranged! Nearest neighbors ≈ 6 eV) → additional new physics??

t : hopping amplitude ( )

)4

( 2

2 0

2

1 r R

R r R e r r d

t=

∫

∑

∑∑

∈

+ + +

−

=

j j j j l l N nl j

j U

t c c cc nn

H ( )

. . ,

σ σ σ σσ

= HBand + HCoulomb

“hopping” “on-site Coulomb repulsion”

Hopping Processes & Hubbard Bands

1. Hopping processes with transition between Hubbard-bands (→change of Coulomb energy):

neutral neutral + -

U

neutral neutral

- +

U

2. Hopping process without transition (same Coulomb-energy):

- neutral neutral -

UHB

+ neutral

neutral +

LHB

3. Forbidden hopping processes:

⇒ in correlated systems, the energy terms for simple hopping processes depend on the occupation of neighboring sites; hopping transports "spin-information"; the apparently simple single electron operator H_bandgets complex many body aspects

upper Hubbard band

lower Hubbard band

E

E

E

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• summary

Cubic Cell a₀ (e. g. CaTiO3)

orthorhombic setting

a ≈b ~ a2 0; c ~ 2 a0 Distorted Perovskites

Sizable octahedral tilts due to misfit of mean ionic radii of A,B ions

→ orthorhombic (LaMnO3Pbnm) or rhombohedral structures, if tolerance factor T ≠ 1:

A,B O

MN O

R R

T 1

R R

2

= ++

A: trivalent cation (A= La, Pr, Nd; Sm; Eu; Gd; Tb, Dy, Ho, Er, Y, Bi) B: divalent cation (B = Sr, Ca, Ba, Pb)

A

B

MnO

[ ][ ]

4 x 3

x 1 2 x 3

x

1

Sr Mn Mn O

La

(ionic model!) Structure: Perovskite related

Example: Mixed Valence Manganites Crystal Field Effect

Loops point between negative charges:

Lower Coulomb energy!

Loops of electron density distribution point towards negative charges:

Coulomb repulsion→ higher energy ! x²-y²

3z²-r²

zx yz xy

Mn ions with 3d orbitals in octahedra of O^2-(“ionic model”)

Jahn-Teller Effekt

d⁴ ≈2 eV

< JH≈4 eV e_g

t_2g

≈0.6 eV

free ion cubic environment

Jahn-Teller distortion

[ ][ ]

4 x 3

x 1 2 x 3

x

1

Sr Mn Mn O

La

Mn³⁺ion:

LaMnO

: Spin and Orbital Order

Below TJT≈780 K:

cooperative Jahn-Teller distortion (minimal macroscopic lattice deform.)

⇒Orbital order

LaMnO3: "d"-type orbital ordering and "A"-type antiferromagnetic ordering result from interplay between structural, orbital and spin degrees of freedom and the relative strength of different coupling mechanisms.

spin order below TN≈145 K:

•Ferromagnetic in a-b planes ("Kugel-Khomskii")

•Antiferromagnetic along c (small overlap of eg- orbitals⇒ AF superexchange of t2gdominates)

J ≈- 10 K J' ≈7 K

CaMnO3: (only t2g⇒ AF exchange) LaMnO3:

Charge-, Orbital- & Spin-Order

Mn³⁺

Mn⁴⁺

O^2-

CE-type charge/orbital in half-doped manganites Mn⁴⁺

Mn³⁺

Example:

Half-doped Manganites

3 2 3 4

1 2 1 2 1 2 1 2 3

La Sr

Mn Mn

O

⎡ ⎤ ⎡ ⎤

⎣ ⎦ ⎣ ⎦

Complex ordering phenomena; subtle interplay between lattice-, charge-, orbital- and spin degrees of freedom; leads to new phenomena like colossal magneto resistance

Magneto-Resistance CMR

Urushibara et al. PRB 51 (1995), 14103

Zero Field Magnetoresistance

Colossal MagnetoResistance (note: 1T ≈0.12 meV≈1.3K)

PMI FMM FMI

Double Exchange

•

FM exchange connected with conductivity

•

t

= t · cos

/

conductivity depends on magnetic order

•But:

Double Exchange: wrong magnitude of resistivity

(Millis et al. PRL 74 (1995), 5144)

→

electron phonon interaction? Zener polarons? …

e_g

J_H

t_2g e_g

J_H

J_AF t

Mn³⁺ Mn⁴⁺

t2g eg

Mn⁴⁺ O^2- Mn³⁺

t2g eg

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• summary

Lattice and Spin Structure

powdered single x-tal

H. Li, Th. Brückel et al.

• ferromagnetic order:

- intensity on top of structural Bragg peaks

• antiferromagnetic order:

- larger unit cell

⇒additional superstructure reflections

• low T-structure:

monoclinic

• structural info

↓

charge and orbital order

↓ CMR-effect

Charge Order – With Neutrons?

“Bond- Valence Sum”: Bond length depends on valence

0 ij

ij

R R

s exp B

⎛ − ⎞

= ⎜ ⎟

⎝ ⎠

with B=0.37 and R₀tabulated for cation-oxygen bonds:

Empirical correlation between chemical bond length and “bond valence”:

The sum of the bond valences around an atom i is (nearly) equal to its valence or oxidation state:

i ij

ij

V=

∑

G.H. Rao, K. Bärner & I.D. Brown J. Phys.: Condens. Matter 10 (1998), L757

Similar: Bond length depends on orbital order

resonant non resonant

→

→orbital order visible in superstructure reflectionsorbital order visible in superstructure reflections εF

E

γ_L

III Templeton & Templeton Acta Cryst. A36 (1980), 436

Anisotropic Anomalous X-Ray Scattering

6.50 6.52 6.54 6.56 6.58 6.60

10⁰ 10¹ 10² 10³ La

7/8Sr

1/8MnO

3 - Resonant Superlattice Ref.

Inorm (cps)

Photon Energy (KeV)

@ 60 K & σ-π (1,0,4.5) (1,0,5.5) (1,0,3.5) (3,0,0.5) (3,0,-0.5)

Orbital Polaron Lattice

• Resonant X-Ray Scattering

x z

y Mn3+

Mn⁴⁺

O^2-

• Lattice of orbital polarons in the ferromagnetic insulating phase of La7/8Sr1/8MnO3(T≤155 K)

Anisotropic anomalous x-ray scattering:

Detailed information on charge- and orbital ordering element specific; combines diffraction and spectroscopy Y. Su, Th. Brückel et al

Quasielastic Scattering T = 170 K magnetic Bragg-peaks T = 120 K

magnetic diffuse

scattering superstructure:

charge- and orbital order

Q_y

La

Sr

MnO

single crystal

Information on magnetic correlations and interactions

Spinwaves in La

Sr

MnO

@ 120K

Q E

Single crystal- TOF-spectrometer yields full information on structure and excitations in one go!

Spinwaves in La0.875Sr0.125MnO3

E

Q_x Q_y

Intensity in 3 /4 Dimensions

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• summary

Complexity in Correlated Electron Systems

charge spin

lattice orbit competing degrees of freedom

High sensitivity

External Fields/

Parameters H E µ T σP d

Complex Collective Behaviour / Novel Ground States CO / OO / SO / JT Spin-Peierls Transition Metal-Insulator Trans.

Cooper Pairs Orbital-/Spin-Liquid

?

Novel functionalities Colos. Magnetores.CMR, High Tc Supercond. HTSC negative thermal exp.

Multiferroica

? Outstanding challenge in condensed matter physics.

Neutron & X-Ray Scattering are indispensable tools to disentangle complexity!