CONTINUOUS CHARGE
MODULATED DIAGONAL PHASE
IN MANGANITES.
cond-mat/0310333 PRL (2004)
Luis Brey
CSIC-Madrid
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: trivalent
D: (Ca, Sr …) divalent
a
La,Ca,Sr, …
x: hole concentration.
Ideal cubic
perovskite structure.
•This structure is distorted (x→0)
Cation size mismatch.
Jahn Teller effects.
•Electric active orbitals Mn
x holes in the Mn d-orbitals
La
Ca
Mn
O
5d1 6s2
4s2
3d54s2
2S2 2p4
3+
2+
3+
2-
SIMPLIFIED PHASE DIAGRAM.
300
200
CO
CO
FMI
FMI
AF
100
AF
Temperature (K)
PMI
0
0.0
FMM
AF
AF
FMM
CO
CAF
CAF
CO
0.2
0.4
0.6
%Ca, x
0.8
1.0
Hwang & Cheong
Application: Colossal Magneto Resistance
La0.75 Ca0.25MnO3
P.Schiffer et al. PRL (’95)
BASIC PHYSICS
Strongly 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…
SIMPLIFIED PHASE DIAGRAM.
300
200
CO
CO
FMI
FMI
AF
100
AF
Temperature (K)
PMI
0
0.0
FMM
AF
AF
FMM
CO
CAF
CAF
CO
0.2
0.4
0.6
%Ca, x
0.8
1.0
Hwang & Cheong
MOTIVATION
Electron microscopy experiments have shown an uniform periodicity
proportional to (1-x) (Loudon et al. cond-mat/0308581)
x=0.5
x=0.52
Previous interpretations:
•AF 1D chains. Uniform charge
x=0.58
x=0.67
Electron diffraction patterns.
•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)
CS
eg
1-x electrons
t2g
3 electrons
4-x electrons per Mn atom
These orbitals can be
treated as an isospin.
|2>=
|1>=
Hund’s Coupling
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
In the xy-plane
t22=4/3 t
t22=t/3
t12=t11=0.
t12x=+t/(3)1/3 t12y= -t/(3)1/3
t11=t
INGREDIENTS.
• Kinetic Energy (two d-orbitals)
• Superexchange interaction between Mn’s.
Superexchange interaction between Mn.
ANTFERROMAGNETIC COUPLING .
eg
Hund’s Coupling
t2g
J AF
∑
rr
Si S j
<i , j >
S=2
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)
•Holes moving around.
JH → ∞
t → t cos
H KE + H Hund = −
f i , j = cos
θ ij
2
e
θ ij
2
•Strong Hund’s coupling. σ·S
e
∑
iφij
•Tunneling conserves spin.
f i , j t aa 'Cia+ C ja '
iφij
Long range ferromagnetic interaction.
INGREDIENTS.
•
•
•
•
Kinetic Energy (two d-orbitals)
Superexchange interaction between Mn’s.
Hunds coupling. (Double Exchange)
Hubbard term.
Hubbard term
Hubbard term U for describing the strong inter-orbital
Coulomb interaction. U penalizes the occupancy of two
orbitals at a site i (orbital ferromagnetism).
H Hub. = U
∑
ni ,a ni ,a '
i ,a ,a '
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.
H el − ph = λ
H elastic
1
=
2
∑ (Q ρ + Q τ
1i i
2i xi
+ Q3iτ zi )
i
∑
( β Q12i + Q22i + Q32i )
i
τ are the orbital pseudospin densities.
τ xi = Ci+1Ci 2 + Ci+2Ci1
τ zi = Ci+1Ci1 − Ci+2Ci 2
Cooperative Jahn-Teller effect
Cooporative. Distortions are inhomogeneous, and produce
long range interactions.
Mn
Oxygen
HAMILTONIAN
H KE + H Hund = −
H el − ph = λ
H elastic
1
=
2
∑
f i , j t aa 'Cia+ C ja '
∑ (Q ρ + Q τ
1i i
2i xi
+ Q3iτ zi )
i
∑
H Hub. = U
( β Q12i + Q22i + Q32i )
i
∑n n
rr
∑S S
i ,a
i ,a '
i ,a ,a '
H AF = J AF
i
<i , j >
j
For a electron density, a given set of
parameters λ, JAF and U, and a texture of
core 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
x=0.5 Phase Diagram (t=1,U=0)
2.0
3D 8x8x8
x=0.5
FM
CO
OO
AM
'Polaronic'
1.5
λ
CE
CO
OO
1.0
CE'
CO
OO
FM
CD
OD
0.5
A
0.0
0.0
0.1
0.2
0.3
JAF
0.4
0.5
CE phase: Charge, spin and orbital order.
•Charge and orbital order stacked
in z-direction.
•Magnetically: AF coupled x-y layers.
Strong
cooperative
distortion ⇒ orbital
orbitals
Dagotto, Khomskii, Millis … Experimentally checked?
Jahn-Teller
order of eg
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.
None of these phases present a
diagonal modulation of the charge.
Alonso et al.
The period does not change
continuously with x.
Dagotto et al
For x=(1-n/m)>0 ,the DIAGONAL PHASE consists of
AF ordered layers composed of AF coupled zig-zag chains,
with vertical and horizontal steps formed by m+1 Mn ions.
• t1,2x = - t 1,2y
•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 charge
modulation of period (a,a)m/n.
•Unit cell 4m×4m. (x=2/3 (3/4), matrix 242 (402))
t
…
m
-t
.
.
.
t
m
-t
PHASE DIAGRAM FOR x=2/3. U=0
Diagonal Phase at x=2/3
red ↑
empty↓
Period of the
charge modulation
(a,a)m/n=(a,a)3
For λ=0.5t and U=4t
n1-n2=0.15
Not Mn3+/Mn4+
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.
In real space the charge
modulation changes
continuously as a/(1-x)
Diagonal Fourier
component of the charge
in the diagonal phase
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 coupled
zig-zag chains with vertical and horizontal steps
formed by m+1 Mn ions.
• Agreement with the electron microscopy
experiments.