In Complex Transition Metal Oxides

Correlated Electrons

In Complex Transition Metal Oxides

Prof. Thomas Brückel FZJ - IFF - Institute for Scattering Methods

& RWTH Aachen - Experimental Physics IV c

School on Pulsed Neutrons -

October 2007 - ICTP Trieste Correlated Electron Systems

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

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

2

0

( ) 1 4 V R e

πε R

= ⋅

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• multiferroics

•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,…) ? extreme many body problem !!

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

• multiferroics

• 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 : Sodium

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 0 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⁺jσcjσ

U : Coulomb repulsion in one orbital: ⁼

∫ ∫

^{Φ − Φ}₋ ⁻

2 1 0

2 2 2 1 2 2

1 4

) ( ) (

r r

R r R r dre 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

=

∫

_πε

∑

∑∑

^{+ ↑ ↓}

∈

+ + +

−

=

j j j j l l N nl j

j U

t cc cc n n

H ( )

. . ,

σ σ σ σσ

= H_Band + H_Coulomb

“hopping” “on-site Coulomb repulsion”

Na⁰ Na⁰ e^-

Na⁺ Na^- ε3s ε3s 0 2ε3s+ U3s hopping t

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

Hubbard-Subbands

single site spectrum in Hubbard-model (occupation-number dependent!):

at+U

ε

εat U

for one electron: for two electrons:

solid:

hopping

t

⇒ level broadened due to hopping into band with width W = z · t (z = number of nearest neighbors) (compare tight binding band theory)

Hubbard bands

E

W = z · t

g (E) W = z · t U

at+U ε

εat

occupation number dependent band structure!

Mott- Hubbard Transition

at half band filling (e.g. CoO)

U = W = z · t

→Metal-insulator- transition

g (E) E

E_F

g (E) E

E_F

U < W

→Metal

g (E) E

E_F

U = 0

→Band metal without correlations E

W = z · t

g (E) W = z · t

U

U >> W = z · t:

Mott-insulator E_F

Correlations lead to band splitting!

For U/t >>1: •away from half-filling →propagating motion of e^-even for U →∞

•half-filling: Mott-Hubbard insulator (see above) In latter case: - all sites are just single occupied

- for t ≠0: virtual hopping between neighbors occurs Pauli-principle: hopping only possible for antiparallel nearest neighbour spins ⇒ antiferromagnetic insulator

t-J and Heisenberg model

t/U

gain in kinetic energy of _U^t 22

−

no hopping can occur (Pauli!)

At half filling, the effective Hamiltonian in the limit U/t>>1 is given by the Heisenberg- Hamiltonian with

for less than half filling, a band term has to be added →H_tJ U

t J

J

N n j i

j i

/ 4²

. . ,

=

⋅

=

∑

∈

S S H

∑

∑

^{+ ⋅}

= ⁺

ij j i ij j i ij

ij

tJ tc c JS S

H _{σ σ}

σ

Separation of low energy spin fluctuations from high energy

charge fluctuations!

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• multiferroics

• 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

1-x

B

x

MnO

3:

[ ][

4

]

3

x 3

x 1 2 x 3

x

1

Sr Mn Mn O

La

₋^{+ +} ₋^{+ +} [ ]^Ar3d4 [ ]^Ar3d3 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

[ ][ ]

3

4 x 3

x 1 2 x 3

x

1

Sr Mn Mn O

La

₋^{+ +} ₋^{+ +} [ ]^Ar3d4 [ ]^Ar3d3

Electron ↔ lattice coupling effect!

Mn

³⁺

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 (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

ij

= t · cos

^ϑij

/

2

→ 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

Mn⁴⁺ O^2- Mn³⁺

t2g

eg

Coupling between different degrees of freedom

Electron-phonon

Jahn Teller active

→Jahn Teller inactive

Electron-orbit Electron↔spin

Mn³⁺ Mn⁴⁺

S = 2 S = ³/₂

S = ³/₂ S = 2 e^- hopping

Double-exchange interaction coupled to conductivity

Charge mobility depends on orbital arrangement!

Charge mobility coupled to phonon bath!

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• multiferroics

• 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=

∑

s

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

La7/8Sr1/8MnO3 - 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 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

see also J. Geck et al

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

magnetic diffuse

scattering superstructure:

charge- and orbital order

Qx

Q_y

La

0.875

Sr

0.125

MnO

3

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

Qx

Qy

Intensity in 3 /4 Dimensions Peculiar spin-wave excitations

0.0 0.1 0.2 0.3 0.4 0.5 0

5 10 15 20 25 30

21.6 meV

15.3 meV 11.2 meV

5.7 meV T = 120 K, H // Q , H = 3.5 T

Energy (meV)

[0, 0, 1+

ξ

]_cubic (r.l.u.)

LO-phonon branch Calculated spin wave dispersion (E = 4 Jeff S [1-cos(2π ξ])

Y. Su, W. Schweika et al - see also Hennion et al

Inelastic neutron scattering:

Modified spin wave dispersion due to complex ordering

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• multiferroics

• summary

Magnetism:

ordering of spins axial vector

Ferroelectricity:

polar arrangement of charges polar vector

PZT Piezoelectricity

Ferromagnets & Ferroelectrics

breaks time inversion symmetry t breaks spatial inversion symmetry i

i

t

- +

i

t

-

+ -

+

requires partially occupied d orbitals requires unoccupied d⁰orbitals mutually

exclusive?

Multiferroics:

materials that combine different ferroic properties, e.g.

materials that are simultaneously ferromagnetic and ferroelectric with a strong coupling between them.

Induction ofpolarization by a magnetic field Induction of magnetization byan electric field

Multiferroics

⇒potential for applications, e.g. in data storage

Cheong et al., Nature Mater. 6, 13 (2006).

Origin of ferroelectricity

Geometric ferroelectrics

Buckling of MnO

5

bipyramids and displacements of Y ions Polarization

van Aken et al., Nature Mater. 3, 164 (2004) Y

MnO5

YMnO

3

P6₃/mmc P6₃cm Hexagonal perovskite

Electronic ferroelectrics

Ikeda et al., Nature. 436, 1136 (2005).

LuFe

₂

O

₄ Charge frustrated system

R-3m

FE magnetic

order

coupling between electric and magnetic degrees of freedom:

Electronic ferroelectrics

Electric polarization was induced by charge ordering in a non-symmetric fashion

Site-centered charge order

Combination of bond-centered and site-centered charge order Efremov et al., Nature Mater. 3, 853 (2004).

Bond-centered charge order

Ferroelectric intermediate state O

TM

Pr

1-x

Ca

x

MnO

3

Magnetic ferroelectrics

TbMnO

3

Inversion symmetry broken

15 K 35 K

TbMn

2

O

5

Strong coupling between ferroelectricty and magnetism Magnetic field induces a sign reversal of the electric polarization.

Kenzelmann et al., PRL. 95, 087206 (2005).

N. Hur et al., Nature. 429, 392 (2004).

Frustrated spin systems

TbMnO ₃

Multiferroic TbMnO

₃

:

orthorhombically distorted perovskite Pbnm layered modulated AF order below 41 K accompanied by lattice modulation FE below 27 K

magnetic field induced electric

polarization flop T. Kimura et al. Nature 426 (2003), 55

Multiferroic TbMnO ₃

T. Kimura et al. Nature 426 (2003), 55

“Giant” coupling between Magnetism (breaking time reversal) and Ferroelectricity (spatial inversion):

in rather small field (> 5T) along b, FE polarization flips from parallel c to parallel a

(would be very interesting for applications e.g. in memory devices, if T_Cwas higher!)

Origin of FE in TbMnO

₃

? - no empty d orbitals - no lone s electron pairs

→ magnetic spiral structure

(breaks inversion symmetry) ?

Magnetic structure TbMnO ₃

M. Kenzelmann et al. PRL 95 (2004), 087206

Neutron diffraction:

→(0,k,1)

Magnetic structure (Mn):

41K > T > 28K:Mn: longitudinally (sinusoidal) modulated AF with propagation vector (0,q,1), q≈0.28; moment direction along b;

Tb: no l.r.o. ;

28K > T > 7K:Mn: elliptical spiral in b-c plane; squaring up (3rd order harmonics) Tb: modulated moment along a;

7K > T: Tb: bunched incommensurate structure (0,t,1), t≈0.425

Excitations inTbMnO ₃

D. Senff et al; PRL 2007

• Crystal field @ 5 meV

• Degeneracy lifted in FE phase

• Mode (1) couples strongly to polarization via DM interaction

coupled magnetic (spin waves) and lattice (phonons) excitations ?

XRES inTbMnO ₃

• Magnetic resonant scattering observed at Tb L

III

edge (also at Tb L

_II

and Mn K)

• strong temperature dependence of magnetic propagation vector

• „quasi“- lock-in at FE transition with strong

hysteresis

J. Voigt et al; PRB 76 (2007), 104431-1

XRES inTbMnO ₃

• energy dependence of resonance shows many features due to complex band structure

• unusual strong enhancement at Mn K edge

• above FE transition: T- dependence desribed by Mn spin moment only: Tb 4f magnetization induced by Mn 3d moment

• below FE transition:

additional Tb 4f ordering

J. Voigt et al; PRB 76 (2007), 104431-1

Strong Coupling of 4f and 3d moments inTbMnO

3 DFT calculation:

partial density of states Magnetization density of 1st unoccupied Mn peak (0 <E − E_F< 2 eV)

for an insulator an unusual strong coupling between the spin polarization in the Mn 3d and Tb 4f bands was revieled by combining XRES and DFT

⇒local mobility of charge carriers?

J. Voigt et al; PRB 76 (2007), 104431-1

Outline

• electronic structure of solids

• electronic correlations

• example: doped manganites – complex ordering phenomena

• experimental techniques:

neutron & x-ray scattering

• multiferroics

• summary

Complexity in Correlated Electron Systems

charge spin

lattice orbit competing degrees of freedom

High sensitivity

External Fields/

Parameters H E µ T 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!