phase separation in strongly correlated electron systems with jahn-teller ions k.i.kugel, a.l....
TRANSCRIPT
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