Spin Hamiltonian and Order out of Coulomb Phase in Pyrochlore - - PowerPoint PPT Presentation

Spin Hamiltonian and Order out of Coulomb Phase in Pyrochlore Structure of FeF3

arXiv: 1407.0849

Farhad Shahbazi in collaboration with Azam Sadeghi (IUT) Mojtaba Alaei (IUT) Michel J. P. Gingras (UWaterloo)

1

Outline

• Experimental observation on Pyr-FeF3
• Derivation of an effective spin Hamiltonian using 


ab initio DFT method

• Monte Carlo Simulation
• Conclusion

2

Experimental Observations

3

Structures of FeF3

G.Ferey et al, Revue de Chimie minerale 23, 474 (1986)

• Rhombohedral (R-FeF3)


• Hexagonal Tungsten Bronze (HTB-

FeF3)
 
 


• Pyrochlore (Pyr- FeF3)


Fe − F − Fe = 142.3 Fe − F − Fe = 152.15 Fe − F − Fe = 141.65

TN = 110K TN = 365K

TN = 20 ± 2K

µ = 4.45µB

µ = 4.07µB

µ = 3.32µB Fe+3 : 3d5 µfree−ion = 5µB

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Pyr-FeF3

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Pyrochlore Structure

• Corner sharing array of

tetrahedra

• Fcc Bravais lattice+ 4

lattice point basis

• In Pyr-FeF3, ions

reside on the corners of the tetrahedra

• The ground state has all-

in/all-out (AIAO) ordering

Fe+3

6

Measurements

• Magnetic Susceptibility

• G. Ferey, et al, Revue de Chimie minerale 23, 474 (1986)



 Results: Deviation from Curie-Weiss law even at T=300K.
 sign of transition at T~20K 


• Mossbauer Study

• Y. Calage, et al, Journal of Solid State Chemistry 69, 197 (1987) 

• Neutron Diffraction


J.N. Reimers, et al, Phys. Rev. B, 5692 (1991);


• Phys. Rev. B 45, 7295 (1992)


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Questions

• Why the transition temperature is too small in Pyr-

FeF3?

• What is the origin of non-coplanar “AIAO” ordering?
• What is the universality class of transition?

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Why the transition temperature is too small in Pyr-FeF3?

• Geometric frustration
• The ground state of nearest

neighbour classical Heisenberg Anti-ferromagnet is highly degenerate on pyrochlore lattice. This model remains disordered down to zero kelvin.
 


• R. Moessner, and J. T Chalker, Phys. Rev

Lett 80, 2929; Phys. Rev. B 58, 12049 (1998)

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What is the origin of non-coplanar “AIAO” ordering?

• Spin anisotropy due to spin-orbit coupling
• But the angular momentum of iron ion is zero, then

where does the spin-obit coupling may come from?

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11

Abinitio DFT Calculation

Microscopic Spin Hamiltonian

Heff = J1 2 X

hi,ji

X

a6=b

na

i · nb j + B

2 X

hi,ji

X

a6=b

(na

i · nb j)2 + D

2 X

hi,ji

X

a6=b

ˆ Dab · (na

i × nb j)

12

13

• M. Elhajal, et al, Phys. Rev. B 71, 094420 (2005)

Direct DM vectors

Energy Landscape of biquadratic term for Single Tetrahedron

• Minimum locates at 

• corresponding to a non-collinear state. 


DM interaction fixes this state to the all-in or all-out directions.
 
 
 The location of the saddle point is
 
 


• corresponding to co-planar states

which have triple degeneracy.
 DM interaction fixes these states to xy, xz or yz planes, depending which two spins are collinear. 
 


Q = X

<i,j>

(Si.Sj)2 = B(1 − 2 sin2 φ cos θ + (3 + cos2 φ) cos2 θ + cos2 φ).

φ = π/2, θ = cos−1(1/3)

/4 /2 3/4

• /4

/2 3/4

• 1

2 3 4 5 6

Q

• Q

1 2 3 4 5 6

φ = 0, π; θ = π/2 φ = π/2; θ = 0

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Coplanar vs AIAO state

EAIAO/N = −J1 + B/3 − 2 √ 2D Ecoplanar/N = −J1 + B − √ 2D

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What is the universality class of transition?

• Monte Carlo simulation 


AIAO order parameter


Order parameter Binder’s cumulant 



 
 Finite size scaling
 
 
 Results
 
 


M = hmiT m = Σi,aSa

i .ˆ

da/N

Um(T) = 1 − 1 3 < m4 > < m2 >2

Tc/J1 = 0.0601(2) β = 0.18(2) ν = 0.60(2) J1 = 32.7eV → Tc =≈ 22K

M = L−β/νM(tL1/ν)

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The critical exponents of specific heat and AIAO susceptibility α + 2β + χ = 2.0(1) α = 0.44(3), χ = 1.20(3)

Deeper Look for the order of transition

• Probability density of the AIAO order

parameter in a tetrahedron


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mn =

4

X

a=1

Sa · da

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• Probability density of Four-spin correlation


R = h(S1 · S2)(S3 · S4) + (S1 · S3)(S2 · S4) + (S1 · S4)(S2 · S3)i

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˜ R ≡ |(S1 · S2)(S3 · S4) − (S1 · S3)(S2 · S4) + (S1 · S4)(S2 · S3)|

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UE(T) ⌘ 1 1 3 hE4i hE2i2

Binder Forth energy cumulant

U min

E

(L) = U ∗ + AL−d + O(L−2d)

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R = 1 2 h 1 − 2 sin2 φ cos θ + (3 + cos2 φ) cos2 θ + cos2 φ i

φ = 0, π; θ = π/2 φ = π/2; θ = 0

Proof of coplanarity above transition temperature R = ˜ R = 1 ⇒ n ˜ R = |1 − sin2 θ(1 + cos φ)|

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Irreducible representations of tetrahedron group

ΛE,1 ≡ 1 √ 3 h (S1 · S2) − 1 2(S1 · S3) − 1 2(S1 · S4) − 1 2(S2 · S3) − 1 2(S2 · S4) + (S3 · S4) i

ΛE,2 ≡ 1 2 h (S1 · S3) − (S1 · S4) − (S2 · S3) + (S2 · S4) i

λGlobal

E

= 4 N "⇣ X

tetra

ΛE,1 ⌘2 + ⇣ X

tetra

ΛE,2 ⌘2 #

λLocal

E

= 4 N "X

tetra

• Λ2

E,1 + Λ2 E,2

• #
• N. Shannon, K. Penc, and Y. Motome, Phys. Rev. B 81, 184409 (2010)

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χLocal

E

= N T h h(λLocal

E

)2i hλLocal

E

i2i

λGlobal

E

= 4 N "⇣ X

tetra

ΛE,1 ⌘2 + ⇣ X

tetra

ΛE,2 ⌘2 #

λLocal

E

= 4 N "X

tetra

• Λ2

E,1 + Λ2 E,2

• #

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Neutron Structure Function

f(q) = h|S⊥(q)|2i S⊥(q) = S S · q/q2 S(q) = X

ri

Si exp(iq · ri)

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0.018 0.2 0.4 0.6 0.8 0 0.015 0.05 0.1 0.15 0.2

J2 J3a

AIAO- Phase Modulated- Phase FeF3 *

The effect of second and third neighbor exchange interactions

The mean field phase diagram Q = 0 Q 6= 0

Conclusion

• An effective spin Hamiltonian containing nearest

neighbour AF Heisenberg, biquadratic and DM interactions, precisely describes the magnetic properties

• f Pyr-FeF3.
• The transition to from disordered to AIAO is weakly first
• rder.
• Possible tricritical or Lifshitz universality class.
• A coulomb phase comprised of short-range coplanar

states is proposed above transition temperature.

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Thanks for your attention

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