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 Farhad Shahbazi in collaboration with Azam Sadeghi (IUT) Mojtaba Alaei (IUT) arXiv: 1407.0849 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) 
 Fe − F − Fe = 142 . 3 � T N = 110 K µ = 4 . 45 µ B Hexagonal Tungsten Bronze (HTB- • FeF3) 
 Fe − F − Fe = 152 . 15 � T N = 365 K µ = 4 . 07 µ B Pyrochlore (Pyr- FeF3) 
 • Fe − F − Fe = 141 . 65 � T N = 20 ± 2 K µ = 3 . 32 µ B Fe +3 : 3 d 5 µ free − ion = 5 µ B 4

Pyr-FeF3 5

Pyrochlore Structure • Corner sharing array of tetrahedra • Fcc Bravais lattice+ 4 lattice point basis • In Pyr-FeF3, ions Fe +3 reside on the corners of the tetrahedra • The ground state has all- in/all-out (AIAO) ordering 6


 
 Measurements • Magnetic Susceptibility 
 G. Ferey, et al , R evue 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) 
 
 7

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? 8


 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) 9

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? 10

Abinitio DFT Calculation 11

Microscopic Spin Hamiltonian H e ff = J 1 j + B j ) 2 + D D ab · ( n a ˆ X X X X X X n a i · n b ( n a i · n b i × n b j ) 2 2 2 h i,j i a 6 = b h i,j i a 6 = b h i,j i a 6 = b 12

Direct DM vectors M. Elhajal, et al , Phys. Rev. B 71, 094420 (2005) 13


 
 
 
 
 Energy Landscape of biquadratic term for Single Tetrahedron ( S i . S j ) 2 = B (1 − 2 sin 2 φ cos θ + (3 + cos 2 φ ) cos 2 θ + cos 2 φ ) . X Q = <i,j> • Minimum locates at 
 φ = π / 2 , θ = cos − 1 (1 / 3) • 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 
 6 6 φ = 0 , π ; θ = π / 2 5 5 4 4 3 φ = π / 2; θ = 0 Q Q 3 2 2 1 1 • corresponding to co-planar states 0 0 which have triple degeneracy. 
 � DM interaction fixes these states to xy, 3 � /4 � xz or yz planes, depending which two 3 � /4 � /2 spins are collinear. 
 � � /2 � /4 � /4 � 0 14

Coplanar vs AIAO state √ E AIAO /N = − J 1 + B/ 3 − 2 2 D √ E coplanar /N = − J 1 + B − 2 D 15


 
 
 
 
 
 
 
 
 
 What is the universality class of transition? • Monte Carlo simulation 
 AIAO order parameter 
 M = h m i T i . ˆ m = Σ i,a S a d a /N Order parameter Binder’s cumulant 
 < m 4 > U m ( T ) = 1 − 1 < m 2 > 2 3 Finite size scaling 
 M = L − β / ν M ( tL 1 / ν ) Results 
 T c /J 1 = 0 . 0601(2) β = 0 . 18(2) ν = 0 . 60(2) J 1 = 32 . 7 eV → T c = ≈ 22 K 16

The critical exponents of specific heat and AIAO susceptibility α = 0 . 44(3) , χ = 1 . 20(3) α + 2 β + χ = 2 . 0(1) 17


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Deeper Look for the order of transition • Probability density of the AIAO order parameter in a tetrahedron 
 4 X S a · d a m n = a =1 18

• Probability density of Four-spin correlation 
 R = h ( S 1 · S 2 )( S 3 · S 4 ) + ( S 1 · S 3 )( S 2 · S 4 ) + ( S 1 · S 4 )( S 2 · S 3 ) i 19

˜ R ≡ | ( S 1 · S 2 )( S 3 · S 4 ) − ( S 1 · S 3 )( S 2 · S 4 ) + ( S 1 · S 4 )( S 2 · S 3 ) | 20

Binder Forth energy cumulant h E 4 i U E ( T ) ⌘ 1 � 1 h E 2 i 2 3 ( L ) = U ∗ + AL − d + O ( L − 2 d ) U min E 21

Proof of coplanarity above transition temperature R = 1 h i 1 − 2 sin 2 φ cos θ + (3 + cos 2 φ ) cos 2 θ + cos 2 φ 2 R = | 1 − sin 2 θ (1 + cos φ ) | ˜ φ = 0 , π ; θ = π / 2 n R = ˜ R = 1 ⇒ φ = π / 2; θ = 0 22

Irreducible representations of tetrahedron group N. Shannon, K. Penc, and Y. Motome, Phys. Rev. B 81, 184409 (2010) 1 ( S 1 · S 2 ) − 1 2( S 1 · S 3 ) − 1 2( S 1 · S 4 ) − 1 2( S 2 · S 3 ) − 1 h i 2( S 2 · S 4 ) + ( S 3 · S 4 ) Λ E , 1 ≡ √ 3 Λ E , 2 ≡ 1 h i ( S 1 · S 3 ) − ( S 1 · S 4 ) − ( S 2 · S 3 ) + ( S 2 · S 4 ) 2 "⇣ X # ⌘ 2 ⌘ 2 = 4 ⇣ X λ Global Λ E , 1 Λ E , 2 + E N tetra tetra "X # = 4 λ Local � Λ 2 E , 1 + Λ 2 � E E , 2 N tetra 23

"⇣ X # = 4 ⌘ 2 ⌘ 2 ⇣ X λ Global Λ E , 1 Λ E , 2 + E N tetra tetra "X # = 4 λ Local Λ 2 E , 1 + Λ 2 � � E E , 2 N tetra = N h i 2 i χ Local h ( λ Local ) 2 i � h λ Local E E E T 24

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Neutron Structure Function f ( q ) = h | S ⊥ ( q ) | 2 i S ⊥ ( q ) = S � S · q / q 2 X S ( q ) = S i exp( i q · r i ) r i 26

The effect of second and third neighbor exchange interactions The mean field phase diagram 0.8 Modulated- Phase 0.6 Q 6 = 0 J 2 0.4 AIAO- Phase Q = 0 0.2 FeF 3 0.018 * 0 0.015 0.05 0.1 0.15 0.2 J 3a 27

Conclusion • An effective spin Hamiltonian containing nearest neighbour AF Heisenberg, biquadratic and DM interactions, precisely describes the magnetic properties of Pyr-FeF3. • The transition to from disordered to AIAO is weakly first order. • Possible tricritical or Lifshitz universality class. • A coulomb phase comprised of short-range coplanar states is proposed above transition temperature. 28

Thanks for your attention 29