Post on 04-Apr-2018
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Double Exchange, Magnetism and Transport inCondensed Matter Physics
J.M.B. Lopes dos Santos1 V.M. Pereira1 E.V. Castro1
A.H. Neto2
1Universidade do Porto & 2Boston University
Advances in Physical Science: meeting in honour of ProfessorAntonio Leite Videira,
U. Aveiro, September, 5-7, 2005
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Outline
1 Double ExchangeExchange MechanismsManganites and DE
2 Ferromagnetic TransitionVariational Mean-FieldRecursion Method
3 Eu1−xCaxB6: A Double Exchange System?Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
4 Summary
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Exchange MechanismsManganites and DE
Origin of Exchange
Magnetic Interactions in Condensed Matter, are hardly ever“properly”magnetic.
CoulombInteractions
Quantum
InteractionsSpin−dependent
Mechanics
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Exchange MechanismsManganites and DE
Origin of Exchange
Magnetic Interactions in Condensed Matter, are hardly ever“properly”magnetic.
CoulombInteractions
Quantum
InteractionsSpin−dependent
Mechanics
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Heisenberg exchange
Coulomb exchange between localized states
J = V ijjiij
Heisenberg exchange is ferromagnetic (Pauli Principle)
−JSi · Sj J > 0
Heisenberg exchange
Coulomb exchange between localized states
J = V ijjiij
Heisenberg exchange is ferromagnetic (Pauli Principle)
−JSi · Sj J > 0
Anderson exchange
Hopping between localized states
J~ t
U
t t
U
2
Anderson exchange is anti-ferromagnetic (Pauli Principle)
−JSi · Sj J < 0
Anderson exchange
Hopping between localized states
J~ t
U
t t
U
2
Anderson exchange is anti-ferromagnetic (Pauli Principle)
−JSi · Sj J < 0
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Exchange MechanismsManganites and DE
Manganites
Chemical formula, AMnO3
a
b
cO2−
T3+ / D2+
Mn3+ / Mn4+
a = b = c = 3.84 ÅE.V. Castro (MSc Thesis, UA) a
b
c
A=La, Nd, Pr, Ca, Sr, Ba, Pb
A2+Mn4+O2−3 → Mn is (3d)3 (x = 1)
A3+Mn3+O2−3 → Mn is (3d)4 (x = 0)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Exchange MechanismsManganites and DE
Manganites
Chemical formula, AMnO3
a
b
cO2−
T3+ / D2+
Mn3+ / Mn4+
a = b = c = 3.84 ÅE.V. Castro (MSc Thesis, UA) a
b
c
A=La, Nd, Pr, Ca, Sr, Ba, Pb
A2+Mn4+O2−3 → Mn is (3d)3 (x = 1)
A3+Mn3+O2−3 → Mn is (3d)4 (x = 0)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Exchange MechanismsManganites and DE
Manganites
Chemical formula, AMnO3
a
b
cO2−
T3+ / D2+
Mn3+ / Mn4+
a = b = c = 3.84 ÅE.V. Castro (MSc Thesis, UA) a
b
c
A=La, Nd, Pr, Ca, Sr, Ba, Pb
A2+Mn4+O2−3 → Mn is (3d)3 (x = 1)
A3+Mn3+O2−3 → Mn is (3d)4 (x = 0)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double Exchange Hamiltonian
Ferromagnetic Kondo model
HFK = −∑i ,j ,σ
tc†iσcjσ − JH
∑i
σ · Si
JH t, S 1 (classical S)
In low energy sector,electrons have spinparallel to Mn spin.
Double Exchange (DE) Model
HDE = −∑
tij [S]c†i cj tij
tij = t 〈θi , ϕi | θj , ϕj〉 = t cos
(θij
2
)eφij
Double Exchange Hamiltonian
Ferromagnetic Kondo model
HFK = −∑i ,j ,σ
tc†iσcjσ − JH
∑i
σ · Si
JH t, S 1 (classical S)
In low energy sector,electrons have spinparallel to Mn spin.
Double Exchange (DE) Model
HDE = −∑
tij [S]c†i cj tij
tij = t 〈θi , ϕi | θj , ϕj〉 = t cos
(θij
2
)eφij
Transport and Magnetism
Jonker and Van Santen (1950)
P.Schiffer et.al (1995);K.H. Kim et al(2002)
Warning
Manganites are much more than DE(orbital degeneracy, phonons, chargeordering, inhomegenities).
Transport and Magnetism
Jonker and Van Santen (1950)
P.Schiffer et.al (1995);K.H. Kim et al(2002)
Warning
Manganites are much more than DE(orbital degeneracy, phonons, chargeordering, inhomegenities).
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Variational Mean-FieldRecursion Method
Effective Spin Hamiltonian
t2t1 t3
S =⇒ Fel[S] =⇒ Eel[S] (T TF)
Fs(T ) = −β−1 lnTrSe−βEel[S]
Mean Field Hamiltonian, Hmf = −h ·∑
i Si
Fmf(M) = 〈Eel〉pm − TSpm(M)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Variational Mean-FieldRecursion Method
Effective Spin Hamiltonian
t2t1 t3
S =⇒ Fel[S] =⇒ Eel[S] (T TF)
Fs(T ) = −β−1 lnTrSe−βEel[S]
Mean Field Hamiltonian, Hmf = −h ·∑
i Si
Fmf(M) = 〈Eel〉pm − TSpm(M)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Variational Mean-FieldRecursion Method
Effective Spin Hamiltonian
t2t1 t3
S =⇒ Fel[S] =⇒ Eel[S] (T TF)
Fs(T ) = −β−1 lnTrSe−βEel[S]
Mean Field Hamiltonian, Hmf = −h ·∑
i Si
Fmf(M) = 〈Eel〉pm − TSpm(M)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Variational Mean-FieldRecursion Method
Effective Spin Hamiltonian
t2t1 t3
S =⇒ Fel[S] =⇒ Eel[S] (T TF)
Fs(T ) = −β−1 lnTrSe−βEel[S]
Mean Field Hamiltonian, Hmf = −h ·∑
i Si
Fmf(M) = 〈Eel〉pm − TSpm(M)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Variational Mean-FieldRecursion Method
Density of States of Disordered Systems.
Lanczos Tridiagonalization
|0〉
b0 |1〉 = H |0〉 − a0 |0〉
a0 = 〈0| H |0〉b0 = 〈1| H |0〉
b1 |2〉 = H |1〉 − a1 |1〉 − b∗0 |0〉
a1 = 〈1| H |1〉b1 = 〈2| H |1〉
...
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Continued Fraction Expansion
H =∑<ij>
tijc†i cj +
∑i
εic†i cj
⇒∑n
an |n〉 〈n|+ bn |n + 1〉 〈n|+ b∗n |n〉 〈n + 1|
+ + ...= +
=
0 0 0 0 0 000
0 0 00
G00(ω) ≡ 〈0| 1
ω − H|0〉 =
1[G 0
00
]−1 − |b0|2 G11
Continued Fraction Expansion
H =∑<ij>
tijc†i cj +
∑i
εic†i cj
⇒∑n
an |n〉 〈n|+ bn |n + 1〉 〈n|+ b∗n |n〉 〈n + 1|
+ + ...= +
=
0 0 0 0 0 000
0 0 00
G00(ω) ≡ 〈0| 1
ω − H|0〉 =
1[G 0
00
]−1 − |b0|2 G11
DOS in PARA and FERRO states.
G00(ω) =1
ω − a0 − |b0|2
ω−a1−|b1|2
ω−a2−...
Convergence of Coefficients
0 10 20 30 40n
1.2
1.4
1.6
1.8
2.0
2.2
b n+1
20×20×2030×30×3040×40×4060×60×60
Density of states
-6 -4 -2 0 2 4 6E
0.0
0.1
0.2
ρ(E)
T = 0T → ∞
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Structure
Eu positions define a cubic lattice
B6: 10 Bonding MO (filled).
Eu2+: SEu = 7/2, a = 4.185 A.
EuB6 is semi-metal,
Band overlapp at X point.
or a (doped) semiconductor?
∆ARPES ∼ 1 eV.
Crystal Structure
Mandrus et al. (PRB, 2003)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Structure
Eu positions define a cubic lattice
B6: 10 Bonding MO (filled).
Eu2+: SEu = 7/2, a = 4.185 A.
EuB6 is semi-metal,
Band overlapp at X point.
or a (doped) semiconductor?
∆ARPES ∼ 1 eV.
Electronic Structure
Massidda et al. (ZPB, 1997)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Structure
Eu positions define a cubic lattice
B6: 10 Bonding MO (filled).
Eu2+: SEu = 7/2, a = 4.185 A.
EuB6 is semi-metal,
Band overlapp at X point.
or a (doped) semiconductor?
∆ARPES ∼ 1 eV.
ARPES measurement
Denlinger et al. (PRL, 2002)
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Ferromagnetic Metal
EuB6 is a ferromagnetic metal.
ρ(T = 0) ≈ 5 µΩ.cm.
TC ≈ 15 K.
MSat.: 7 µB /Eu
Small carrier density (B6 vacancies)
ne(T > TC ) ≈ 0.003 per unit cell.
ne(T > TC ) ≈ 0.009 per unit cell.
Carrier density highly enhanced below TC
DC resistivity: ρ(T ) adm M
Paschen et al (PRB, 2000)
Ferromagnetic Metal
EuB6 is a ferromagnetic metal.
ρ(T = 0) ≈ 5 µΩ.cm.
TC ≈ 15 K.
MSat.: 7 µB /Eu
Small carrier density (B6 vacancies)
ne(T > TC ) ≈ 0.003 per unit cell.
ne(T > TC ) ≈ 0.009 per unit cell.
Carrier density highly enhanced below TC
Carrier Density
Paschen et al (PRB, 2000)
Ferromagnetic Metal
EuB6 is a ferromagnetic metal.
ρ(T = 0) ≈ 5 µΩ.cm.
TC ≈ 15 K.
MSat.: 7 µB /Eu
Small carrier density (B6 vacancies)
ne(T > TC ) ≈ 0.003 per unit cell.
ne(T > TC ) ≈ 0.009 per unit cell.
Carrier density highly enhanced below TC
Carrier Density
Paschen et al (PRB, 2000)
Optical Response
Blue shift of plasma frequency at TC :
FM enhances ωP
Broderick et al (PRB 2002)
Scaling of ωP with M
Broderick et al (PRB 2002)
Plasma frequency scales with the magnetization:
ωP(T ,H) = ωP(M)
To insulator (upon doping).
Increasing the Ca content:
ρ(T ) evolves to an insulating behavior(T > TC )
TC decreases with doping strenght x .
ne is (more) enhanced below TC .
A mobility gap?
ρ(T ) scales exponentially M(T ).
ωP also scales exponentially withM(T ).
ρ(T ) at 40% doping
Wigger et al. (PRB 2002)
To insulator (upon doping).
Increasing the Ca content:
ρ(T ) evolves to an insulating behavior(T > TC )
TC decreases with doping strenght x .
ne is (more) enhanced below TC .
A mobility gap?
ρ(T ) scales exponentially M(T ).
ωP also scales exponentially withM(T ).
Gap-like behavior
Peruchi et al. (PRL 2004)
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Disorder and localization in the DEM
What can DE model provide?
N(E ,M) ≡⟨N(E , ~Si)
⟩pm
(Recursion Method)
M dependent FL, EF(M)
M dependent mobility
edge,EC(M) =⟨EC(E , ~Si)
⟩(Transfer Matrix)
EC . EF for ne ∼ 10−3. The down shift of EC(M) enhances ne !
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Disorder and localization in the DEM
What can DE model provide?
N(E ,M) ≡⟨N(E , ~Si)
⟩pm
(Recursion Method)
M dependent FL, EF(M)
M dependent mobility
edge,EC(M) =⟨EC(E , ~Si)
⟩(Transfer Matrix)
Mobility Edge
N(E)
EE FC
E
EC . EF for ne ∼ 10−3. The down shift of EC(M) enhances ne !
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Disorder and localization in the DEM
What can DE model provide?
N(E ,M) ≡⟨N(E , ~Si)
⟩pm
(Recursion Method)
M dependent FL, EF(M)
M dependent mobility
edge,EC(M) =⟨EC(E , ~Si)
⟩(Transfer Matrix)
Results for EC
0 0.2 0.4 0.6 0.8 1M / MS
-6
-5.5
-5
-4.5
-4
-3.5
E c / t
Mobility edge and Mobility Gap
0
0.1
0.2
0.3
0.4
0.5
0.6
- ∆ /
t
(Bottom of the band is at -6 in the left axis).
∆(M, ne ) = EF(M)− EC(M, ne ).
EC . EF for ne ∼ 10−3. The down shift of EC(M) enhances ne !
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Disorder and localization in the DEM
What can DE model provide?
N(E ,M) ≡⟨N(E , ~Si)
⟩pm
(Recursion Method)
M dependent FL, EF(M)
M dependent mobility
edge,EC(M) =⟨EC(E , ~Si)
⟩(Transfer Matrix)
Results for EC
0 0.2 0.4 0.6 0.8 1M / MS
-6
-5.5
-5
-4.5
-4
-3.5
E c / t
Mobility edge and Mobility Gap
0
0.1
0.2
0.3
0.4
0.5
0.6
- ∆ /
t
(Bottom of the band is at -6 in the left axis).
∆(M, ne ) = EF(M)− EC(M, ne ).
EC . EF for ne ∼ 10−3. The down shift of EC(M) enhances ne !
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Results for the transport properties - EuB6
Carrier density
0 5 10 15Temperature (K)
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
n e (ext
ende
d)
Extended carrier densityEw = 0.1 t , ne = 0.003
Only EF is tuned for T > TC.
Plasma frequency
5.0×106 1.0×107 1.5×107 2.0×107 2.5×107
ωp2 (cm-2)
0
0.2
0.4
0.6
0.8
1
M /
MS
Plasma FrequencyPure diagonal, ne(M=0) = 0.007, t = 0.7 eV
ω2P ∝ 〈H〉 =
tR∞
ECN(ε,M)f (ε)εdε.
These (and other) results reproduce the experimental signatures ofEuB6!
Ca Doping and percolation
Doping dilutes magnetic sites and hopping in the lattice: expect mobility edgeto move into band.
Eu1 − xCaxB6
FIFM
PM
T (K
)
15
x MIP x MI cx pc
cT (x)
0 1
PI
Proposed Phase diagram Wigger et al., (PRL, 2004).
We expect Eu0.60Ca0.40B6 to lie between xPMI and xF
MI. Above TC it has amobility gap.
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Independent Polaron
By creating a ferromagnetic bubble, an electron lowers its energy; but there is amagnetic entropy cost.
∆Fpol(R, T )/ne =4t − 6t cos(π/(R + 1)) +TR3 log(2S + 1) − TScfg
Minimization gives Req(T ),and stability temperature, Tm.
Percolation argumentestimates TC.
Electronic wavefunction of Polaron.
Polarons are seen in Raman scattering at the predicted temperatures. TC isdepressed relative to mean-field estimate.
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Experimental signaturesThe Double Exchange ModelPolarons and Phase Separation
Independent Polaron
By creating a ferromagnetic bubble, an electron lowers its energy; but there is amagnetic entropy cost.
∆Fpol(R, T )/ne =4t − 6t cos(π/(R + 1)) +TR3 log(2S + 1) − TScfg
Minimization gives Req(T ),and stability temperature, Tm.
Percolation argumentestimates TC.
Electronic wavefunction of Polaron.
Polarons are seen in Raman scattering at the predicted temperatures. TC isdepressed relative to mean-field estimate.
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Phase separation
At low densities system does not support homogeneous phase!
Below TC(n), there are twophases of different densities,n+, n−
In GCE (constant µ) transitionis first order, M(TC) 6= 0,∆n 6= 0,
Phase diagram at low ne .
PS region much larger than polaron region
Where have all the polarons gone?.
Phase separation
At low densities system does not support homogeneous phase!
Below TC(n), there are twophases of different densities,n+, n−
In GCE (constant µ) transitionis first order, M(TC) 6= 0,∆n 6= 0,
Phase diagram at low ne .
PS region much larger than polaron region
Where have all the polarons gone?.
Coulomb Supression of Phase Separation
Electrons are charged!
Electrostacic energy(x = V+/V−)
Eel
V=
2
5e2πn2R2 2 + x − 3x1/2
x
Energy of localization
Eloc
E∞(n)=
1 +
15
16
“ π
6n
”1/3 x1/3
R
!
Equilibrium radius ∼ Rpol
Electrostatic supression of PS.
PS region shrinks to polaron region
Polarons are back!
Coulomb Supression of Phase Separation
Electrons are charged!
Electrostacic energy(x = V+/V−)
Eel
V=
2
5e2πn2R2 2 + x − 3x1/2
x
Energy of localization
Eloc
E∞(n)=
1 +
15
16
“ π
6n
”1/3 x1/3
R
!
Equilibrium radius ∼ Rpol
Electrostatic supression of PS.
PS region shrinks to polaron region
Polarons are back!
Controversy
Does Double Exchange apply when J . t?
Yes, at very low densities.
Low energy states have somepolarization along local Mnspin direction, even at M = 0.
In weak coupling one recoversTRKKY
C from DE picture.
t
J
E
N(E)
DE is good starting point for intermediate coupling.
At very low density, many of the concepts (M dependent mobility edge) gothrough, even in weak coupling RKKY limit.
Controversy
Does Double Exchange apply when J . t?
Yes, at very low densities.
Low energy states have somepolarization along local Mnspin direction, even at M = 0.
In weak coupling one recoversTRKKY
C from DE picture.
t
J
E
N(E)
DE is good starting point for intermediate coupling.
At very low density, many of the concepts (M dependent mobility edge) gothrough, even in weak coupling RKKY limit.
Controversy
Does Double Exchange apply when J . t?
Yes, at very low densities.
Low energy states have somepolarization along local Mnspin direction, even at M = 0.
In weak coupling one recoversTRKKY
C from DE picture.
t
J
E
N(E)
DE is good starting point for intermediate coupling.
At very low density, many of the concepts (M dependent mobility edge) gothrough, even in weak coupling RKKY limit.
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Summary
DE for Double Exchange or Deceptively Easy?
The Eu1−xCaxB6 seems to be a cleaner (if unexpected) caseof a DE system.
Simple ideas still have (some) room in Condensed MatterTheory.
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Summary
DE for Double Exchange or Deceptively Easy?
The Eu1−xCaxB6 seems to be a cleaner (if unexpected) caseof a DE system.
Simple ideas still have (some) room in Condensed MatterTheory.
J. Lopes dos Santos et. al. Universidade do Porto & Boston University
Double ExchangeFerromagnetic Transition
Eu1−xCaxB6: A Double Exchange System?Summary
Summary
DE for Double Exchange or Deceptively Easy?
The Eu1−xCaxB6 seems to be a cleaner (if unexpected) caseof a DE system.
Simple ideas still have (some) room in Condensed MatterTheory.
J. Lopes dos Santos et. al. Universidade do Porto & Boston University