In Complex Transition Metal Oxides

Correlated Electrons

In Complex Transition Metal Oxides

Prof. Thomas Brückel

IFF - Institute for Scattering Methods

& RWTH Aachen - Experimental Physics IV c

Neutron Laboratory Course 2009

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

corelevel 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] 3d⁷ 4s²

O: [He] 2s² 2p⁴

⇒ 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 ≈ 10⁸ Ω cm @ RT; compare: Fe → ρ ≈ 10^-7 Ω cm) 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 = 1s² 2s² 2p⁶ 3s¹

ok (ρ ≈ 5·10^-6 Ω cm ) but should hold for a → ∞

3s-band is half filled ⇒ Na ≡ metal

according to Heisenberg Δ ⋅ Δ ≥p x h/ 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 transition from metal to insulator for a critical value of a

Na⁰ Na⁰ e^-

Na⁺ Na^-

ε_3s ε_3s O 2ε_{3s +} U_3s

single particle

energy for3s electron

intraatomicCoulomb 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 −R_j)σ occupation operator c⁺jσcjσ

U : Coulomb repulsion in one orbital: ⁼

∫ ∫

2 0 1

2 2

2 1

2 2

1 4

) (

) (

r r

R r

R r dr e

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 Φ −

− − Φ

=

∫

∑

∑∑

∈

+ + +

−

=

j

j j j

l l

N n

l j

j U

t c c c c n n

H ( )

. ,.

σ σ σ σ σ

= H_Band + H_Coulomb

“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_band gets 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 (LaMnO₃ Pbnm) 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

[ ^La

^Sr

][ ^Mn

^Mn

] ^O

[ ]

[ ]

Charge neutrality → mixed valence Manganese (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 !

3z²-r²

zx yz xy

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

x²-y²

Jahn-Teller Effekt

d⁴

≈2 eV

< J_H≈4 eV e_g

t_2g

≈0.6 eV

free ion cubic

environment Jahn-Teller distortion

[ ^La

^Sr

][ ^Mn

^Mn

] ^O

[ ]

[ ]

Electron ↔ lattice coupling effect!

Mn

ion:

LaMnO ₃ : Spin and Orbital Order

Below T_JT ≈ 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 T_N ≈ 145 K:

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

• Antiferromagnetic along c ^{(smalloverlap of e}_g^-

orbitals⇒AF superexchange of t_2gdominates)

J ≈ - 10 K

J' ≈ 7 K

CaMnO₃:

(only t_2g⇒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? …

t_2g e_g

J_H

t_2g e_g

J_H

J_AF t

Mn³⁺ Mn⁴⁺

t_2g e_g

Mn^{t2g 4+} O^2- Mn³⁺

e_g

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

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

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 R₀ tabulated for cation-oxygen bonds:

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

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

Charge Order – With Neutrons?

“Bond- Valence Sum”

0 ij

ij

R R

s exp

B

⎛ − ⎞

= ⎜ ⎟

⎝ ⎠

with B=0.37 and R₀ tabulated 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

Fe

OBO

[Angst et al., PRL 99, 086403 (2007); 256402 (2007)]

Fe^2.5+ Fe^2.5+ Fe^2.5+ Fe^2.5+

O O O

Fe^2.5+ Fe^2.5+ Fe^2.5+ Fe^2.5+

O O O

0 2 4 6

-2 -1 0

000 020 110 k =

h =

355 K

Fe³⁺ Fe²⁺ Fe³⁺ Fe²⁺

O O O

Fe³⁺ Fe²⁺ Fe³⁺ Fe²⁺

O O O

100 K

2.0 2.2 2.4 2.6 2.8 3.0

100 K

EuBaFe₂O₅

100 K 295 K

Fe₃O₄ Fe₂OBO₃

Fe2a Fe1a Fe2b Fe1b

90 K

Bond-Valence-Sum

resonant non resonant

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

ε_F

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_1/8MnO₃ - Resonant Superlattice Ref.

I norm (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 La_7/8Sr_1/8MnO₃ (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_x Q_y

La

Sr

MnO

single crystal

Information on magnetic correlations and interactions

Spinwaves in La _0.875 Sr _0.125 MnO ₃

@ 120K

Q E

Single crystal- TOF-spectrometer yields full information

on structure and excitations in one go!

Spinwaves in La_0.875Sr_0.125MnO₃

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

HE µT Pσ 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!