CONTINUOUS CHARGE MODULATED DIAGONAL PHASE

IN MANGANITES.

Luis Brey CSIC-Madrid

cond-mat/0310333 PRL (2004)

OUTLINE• MANGANITES.

-Applications.

-Basic Research: Orbital, spin and charge ordering. Phase separation. Strong correlated system.

• GROUND STATE, for doping x>0.5.

-Motivation.

-Ingredients and model.

-Phase Diagram, Diagonal phases.

La1-xDxMnO3 Mn

Oxygen

La,Ca,Sr, …

La: trivalent

D: (Ca, Sr …) divalent

x: hole concentration.

a

Ideal cubic perovskite structure.

•This structure is distorted (x→0)Cation size mismatch.Jahn Teller effects.

•Electric active orbitals Mnx holes in the Mn d-orbitals

La 5d1 6s2 3+Ca 4s2 2+Mn 3d54s2 3+O 2S2 2p4 2-

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

AF

PMI

CAF

COA

F

FMI

FMMCO

Te

mpe

ratu

re (K

)

%Ca, x

FMMFMI

CO

CO

AF CAF

AF

SIMPLIFIED PHASE DIAGRAM.

Hwang & Cheong

Application: Colossal Magneto Resistance

P.Schiffer et al. PRL (’95)

La0.75 Ca0.25MnO3

BASIC PHYSICSStrongly correlated system.

• AF coupling between Mn ion spins.• Kinetic Energy• Coulombic repulsion• Electron phonon coupling.• Temperature

In manganites the energies involved in these interactions are comparable so different ground state can have very similar energies.

• Ferromagnetic metallic phase• AF Mott insulator• Stripe phases• Ferromagnetic charge ordered phases• Phase separation…

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

AF

PMI

CAF

COA

F

FMI

FMMCO

Te

mpe

ratu

re (K

)

%Ca, x

FMMFMI

CO

CO

AF CAF

AF

SIMPLIFIED PHASE DIAGRAM.

Hwang & Cheong

x=0.5 x=0.52

x=0.58 x=0.67

Electron diffraction patterns.

MOTIVATION

Electron microscopy experiments have shown an uniform periodicity

proportional to (1-x) (Loudon et al. cond-mat/0308581)

Previous interpretations:•AF 1D chains. Uniform charge

•Commensurate regions of density x=(1-1/n) , n=2,3… separated by solitons.

INGREDIENTS.

• Kinetic Energy (two d-orbitals)

KINETIC ENERGY La1-xCaxMnO3

●Active orbitals Mn d (5 plus spin) 4-x electrons per Mn atom

CS

eg

t2g

Hund’s Coupling|2>= |1>=

These orbitals can be treated as an isospin.

A density of holes, x, moving in these orbitals. The hopping is through the oxygen, and depends on the orbital type and on the hopping direction.

In the z-direction

t22=4/3 t

t12=t11=0.

In the xy-plane

t22=t/3 t12

x=+t/(3)1/3 t12y= -t/(3)1/3

t11=t

3 electrons

1-x electrons

INGREDIENTS.

• Kinetic Energy (two d-orbitals)• Superexchange interaction between Mn’s.

Superexchange interaction between Mn.

ANTFERROMAGNETIC COUPLING .

Hund’s Coupling

S=2

t2g

eg

jji

iAF SSJrr∑

>< ,

Classical spins.

INGREDIENTS.

• Kinetic Energy (two d-orbitals)• Superexchange interaction between Mn’s.• Hunds coupling. (Double Exchange)

Hund’s coupling. Double Exchange Mechanism.(Zener, DeGennes, Anderson, ‘50)

ijiij

H

ett

Jφθ

2cos→

∞→•Holes moving around.

•Strong Hund’s coupling. σ·S•Tunneling conserves spin.

Long range ferromagnetic interaction.

ijiijji

jaiaaajiHundKE

ef

CCtfHH

φθ2

cos,

'',

=

−=+ +∑

INGREDIENTS.

• Kinetic Energy (two d-orbitals)• Superexchange interaction between Mn’s.• Hunds coupling. (Double Exchange)• Hubbard term.

Hubbard termHubbard term U for describing the strong inter-orbital Coulomb interaction. U penalizes the occupancy of two orbitals at a site i (orbital ferromagnetism).

∑=',,

',,.aai

aiaiHub nnUH

ni,a : occupation of orbital a at site i.

INGREDIENTS.

• Kinetic Energy (two d-orbitals)• Superexchange interaction between Mn’s.• Hunds coupling. (Double Exchange)• Hubbard term.• Electron phonon coupling.

Electron Phonon Coupling.

The active Jahn-Teller modes of the oxygen octahedra, couples with the eg orbitals.

( )

)(21 2

322

21

321

iiii

elastic

iziixiiiiphel

QQQH

QQQH

++=

++=

∑

∑−

β

ττρλ

τ are the orbital pseudospin densities.2211

1221

iiiizi

iiiixi

CCCCCCCC

++

++

−=+=

ττ

Cooperative Jahn-Teller effect

Mn

Oxygen

Cooporative. Distortions are inhomogeneous, and produce

long range interactions.

HAMILTONIAN

'', jaiaaajiHundKE CCtfHH +∑−=+

( )

)(21 2

322

21

321

iiii

elastic

iziixiiiiphel

QQQH

QQQH

++=

++=

∑

∑−

β

ττρλ

∑=',,

',,.aai

aiaiHub nnUH

jji

iAFAF SSJHrr∑

><

=,

For a electron density, a given set of parameters λ, JAF and U, and a texture ofcore spins {Si} we solve self-consistently the mean field version of the Hamiltonian and obtain the energy, the local charges, and the orbital orientation. We have solved the model for hole densities x<0.5 and we have recovered the GS proposed previously. Therefore we can proceed to study x>0.5

3D 8x8x8 x=0.5

JAF0.0 0.1 0.2 0.3 0.4 0.5

λ

0.0

0.5

1.0

1.5

2.0

FMCDOD

CECOOO

AM'Polaronic'

FMCOOO

CE'COOO

A

x=0.5 Phase Diagram (t=1,U=0)

CE phase: Charge, spin and orbital order.•Charge and orbital order stacked in z-direction.•Magnetically: AF coupled x-y layers.

Strong cooperativeStrong cooperative Jahn-Teller distortion ⇒ orbital order of eg orbitals

Dagotto, Khomskii, Millis … Experimentally checked?

For x>0.5 we have studied different magnetic textures…

• 3D FM• 2D FM planes coupled AF (A-phase)• 1D FM chains coupled AF (C-phase)• CE-phase• Island phases• Skyrmion phases• CxE1-x phases• …

Aliaga et al.

Alonso et al.

Dagotto et al

None of these phases present a

diagonal modulation of the charge.

The period does not change

continuously with x.

For x=(1-n/m)>0 ,the DIAGONAL PHASE consists ofAF ordered layers composed of AF coupled zig-zag chains, with vertical and horizontal steps formed by m+1 Mn ions.

m

…

.

.

.m

t t -t

-t• t1,2x = - t 1,2

y

•Modulation with period 2m•Charge Fourier component (2π/a, 2π/a)m/n. •Gaps in the spectra at x=1-n/m (band insulator)•Interactions (U and λ) enlarge the chargemodulation of period (a,a)m/n.•Unit cell 4m×4m. (x=2/3 (3/4), matrix 242 (402))

PHASE DIAGRAM FOR x=2/3. U=0

Diagonal Phase at x=2/3

Period of thecharge modulation(a,a)m/n=(a,a)3

For λ=0.5t and U=4t

n1-n2=0.15

Not Mn3+/Mn4+

red ↑

empty↓

λ-U PHASE DIAGRAM

For x →0.5, m is very large. For λ=0, the diagonal phase becomes unidimensional, and larger unit cells can be studied.

Diagonal Fourier component of the charge in the diagonal phase

In real space the charge modulation changes continuously as a/(1-x)

SUMMARY• We propose a diagonal phase for manganites at

doping x=(1-n/m)>0.5. In this phase the charge is modulated diagonally with a prevalent Fourier component m/n(2π/a,2π/a).

• Magnetically the phase consists of AF coupledzig-zag chains with vertical and horizontal stepsformed by m+1 Mn ions.

• Agreement with the electron microscopyexperiments.