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

Mn Ion

t2g

eg

XY

Z

d3 z2 - r2

XY

Z

dx2 - y2

XY

Z

dxy

XY

Z

dyz

XY

Z

dxz

3d

Mn 4+ Mn 3+

Double Exchange

tMn−O

tMn−O

Mn 4+

O2−

O−

Mn 4+

O2−

Mn 3+

Mn 3+Mn 3+

Mn 3+

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