Phase separation in strongly correlated electron systems

with Jahn-Teller ions

K.I .Kugel, A.L. Rakhmanov, and A.O. Sboychakov

Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences,

Izhorskaya Str. 13/19, Moscow, 125412 Russia

OUTLINEPhase separation and the doping level Interplay between the localization and

metallicity: a minimal model Properties of homogeneous states: FM

metallic and AFM insulatingInhomogeneous states: FM-AFM phase

separation etc.Phase diagram in the temperature-doping

planeEffects of applied magnetic fieldConclusions

Phase separation and the doping levelThe phase separation phenomena: a key issue in the physics of strongly correlated electron systems, especially in manganites and related compounds

The simplest type: formation of nanoscale inhomogeneities such asferromagnetic metallic droplets (magnetic polarons or ferrons) located in an insulating antiferromagnetic matrix self-trapping of charge carriers

What charge carries are self-trapped?If all, it leads to unphysical results

The doping level and the number of self-trapped carriers can differ drastically

Possible cause:The competition between the localization induced bylattice distortions due to the Jahn-Teller effect (orbital ordering)and the gain in kinetic energy due to intersite hopping of charge carriers

The aim: To analyze a simplified model of such competition

Mn3+ (Jahn-Teller ion) Mn4+

dx2-y

2 – stretching of the octahedron in xy plane

d3z2-r

2 - stretching of the octahedron along z axis

As a result, the Jahn-Teller gap EJT arises

Electron structure of Mn ions

Electron Hamiltonian: general form

el-elJTAFMel HHHHH

mn,

mnAFM SSJH

mn,

'n',nnmn,

nbnnmel )σ(2

..a,b,σ

aaH

a,b,σa

ab aaJ

chaatH S

n n

23n

22nn3n2

,,JT QQ

2)()(

KaQQagH nbab

zab

x

bana

',,,n

'nn2

,,nnn

1elel 22

a

aaa

aa nnU

nnU

H

Effective Hamiltonian: localized and itinerant charge carriers

nnnnm

mn,

2

nmJT

nm

mn,nmmneff cos

2coscc bll nnUJSncctH

HJ )2/cos(tteff is the canting angle

Two groups of electrons: localized, l, at JT distortions an itinerant (band), b

n

nneff'eff bl nnHH

JT is the JT energy gain for l electrons counted off the bottom of b-electron band

is the chemical potential

Similar model: T.V. Ramakrishnan et al., PRL 92, 157203 (2004)

Analysis of the effective Hamiltonian

Mean field approach

)()1(

1),(

kk

l

lb n

nG

xnn lb 1Gives nb and nl

Hubbard I type decoupling at fixed magnetic structure

)2/cos(1

2exp

/

2

00

Tett

Green function for b electrons (U/t>>1):

)1(2 lntzW Band width depends on nl

polaronic band narrowing

Densities of localized and itinerant electrons

xnnn lb 1

JT0

)1(2 lntzW

x2<x<1x1<x<x20<x<x1

nl=0

n b=0

nb≠0, nl≠0

Comparison of energies at T=0

Ferromagnetic (FM): magnJTkinFM EEEE

3/2

3/220kin 1

)6(53

1)1(l

bbl n

n

znwnE

lnE JTJT 2magn zJSE 00 ztw

2JTAF )1( zJSxE

xnn lb 1

0bnAntiferromagnetic (AF):

Energies of ferro- and antiferromagnetic states

0.0 0.2 0.4 0.6 0.8 1.0

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

EAF

EFM

n b=0

x2=0.543

EF

M/w

0, E

AF/w

0

xx1=0.05

nl=0

05.0/ 0JT w 005.0/ 02 wzJS

Homogeneous states at T0

2. Ferromagnetic (FM):

0bn1. Antiferromagnetic (AF): JTN ~

,0ln 0bn

blc nntzT )1(~ )2/cos(1

2exp

/

2

00

Tett

3. Canted: ,0ln 1bnfavorable near x=1

4. Paramagnetic (PM): (a) 0bn

0bn(b) with the growth of T transforms to PM with

0

Free energies for homogeneous states

F= min (FFM,FAF,FPM, FCant)

Phase diagram for homogeneous states

Inhomogeneous states (FM-AF phase separation)

AF

insulator

0ln

FM metal0ln

0bn

FM metal

Spatial separation of localized and itinerant electrons is favorable in energy

What determines the size of inhomogeneities?

Energy of inhomogeneous state

sizedroplet determines

surfCoulAFFMinhom )1( EEEppEE

p is the content of FM phase

ECoul is the Coulomb energy related to inhomogeneous charge distribution

Esurf is surface energy of the droplets

EFM is the minimum energy of ferromagnetic phase (at x=x2, nl=0)

EAF is the minimum energy of antiferromagnetic phase (at x=0, nb=0)

Coulomb energy

pppdR

xxVE af

3/1

22

0Coul 32)(5

2

,2

0 de

V

xf and xa are the densities of charge carriers in FM and AF phases, respectively. Here xf =x2, xa=0.

d is lattice constant, is average permittivity

R is the droplet radius

Spherical model: Wigner-Seitz approximation

Each droplet is surrounded by spherical layer of the opposite charge.

Surface energy

5.0 ),(3

surf pxRd

pE f

)( fx is surface tension of metallic droplet calculated using bulk density of states with size-effect corrections

Minimization of the sum ECoul~R2 and Esurf~1/R gives size R of the droplet

3/1

3/1220 )32()(4

15

ppxxVdR

af

3/4~)( bf ntx

Radius of a droplet

0.0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1.0

1.2

1.4

R(x

)/d

x

FMAF m

atrix

FM

AFFM

mat

rix

AF

0.15/ 00 wV

Energy of inhomogeneous state

0.0 0.1 0.2 0.3 0.4 0.5

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04 54

32

FM

Ene

rgy

x

AF

1

Dashed lines correspond to the energies of phase-separated state at different values of parameter V0/w0: 1 - 0, 2 – 1/4, 3 – 1, 4 – 3/2, 5 – 2.

Phase diagram including inhomogeneous states

1. – homogeneous AF state (nb=0)2. – PM(nb0) – PM(nb=0) phase-separated state

Effect of applied magnetic fieldThe most interesting situation – near the transition from PS to a homogeneous state.

At the transition, the density of b electrons undergoes a jump nb

0

The magnetic field shifts the transition point: TPS=TPS(H) => a jump in the relative change of electron density

)0(

)0()()(

b

bbb n

nHnHn

SUMMARY

A “minimal” model dealing with the competition between the localization and metallicity in manganites was formulated It is demonstrated that the number of itinerant charge carriers can be significantly lower than that implied by the doping level. A strong tendency to the phase separation was revealed for a wide doping range