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

Transcription

1 Spin Hamiltonian and Order out of Coulomb Phase in Pyrochlore Structure of FeF3 Farhad Shahbazi in collaboration with Azam Sadeghi (IUT) Mojtaba Alaei (IUT) Michel J. P. Gingras (UWaterloo) arxiv:

2 Outline Experimental observation on Pyr-FeF3 Derivation of an effective spin Hamiltonian using ab initio DFT method Monte Carlo Simulation Conclusion 2

3 Experimental Observations 3

4 Structures of FeF3 G.Ferey et al, Revue de Chimie minerale 23, 474 (1986) Rhombohedral (R-FeF3) Fe F Fe =142.3 T N =110K µ =4.45µ B Hexagonal Tungsten Bronze (HTB- FeF3) Fe F Fe = T N =365K µ =4.07µ B Pyrochlore (Pyr- FeF3) Fe F Fe = T N =20± 2K µ =3.32µ B Fe +3 :3d 5 µ free ion =5µ B 4

5 Pyr-FeF3 5

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

7 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) 7

8 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

9 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, (1998) 9

10 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

11 Abinitio DFT Calculation 11

12 Microscopic Spin Hamiltonian H e = J 1 2 X hi,ji X a6=b n a i n b j + B 2 X hi,ji X (n a i n b j) 2 + D 2 a6=b X hi,ji X a6=b ˆD ab (n a i n b j) 12

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

14 Energy Landscape of biquadratic term for Single Tetrahedron Q = X <i,j> (S i.s j ) 2 = B(1 2sin 2 cos + (3 + cos 2 )cos 2 + cos 2 ). Minimum locates at = /2, =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 =0, ; = /2 = /2; =0 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 /4 /2 /4 0 /4 /2 3 /

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

16 What is the universality class of transition? Monte Carlo simulation AIAO order parameter Order parameter Binder s cumulant Finite size scaling Results M = hmi T m = i,a S a i.ˆd a /N U m (T )=1 1 3 T c /J 1 =0.0601(2) =0.18(2) =0.60(2) <m 4 > <m 2 > 2 M = L / M(tL 1/ ) J 1 =32.7eV! T c = 22K 16

17 The critical exponents of specific heat and AIAO susceptibility =0.44(3), =1.20(3) +2 + =2.0(1) 17

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

19 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

20 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

21 Binder Forth energy cumulant U E (T ) he 4 i he 2 i 2 U min E (L) =U + AL d + O(L 2d ) 21

22 Proof of coplanarity above transition temperature R = 1 2 h1 2sin 2 cos + (3 + cos 2 )cos 2 + cos 2 i R = 1 sin 2 (1 + cos ) R = R =1) n =0, ; = /2 = /2; =0 22

23 Irreducible representations of tetrahedron group N. Shannon, K. Penc, and Y. Motome, Phys. Rev. B 81, (2010) E,1 1 p 3 h(s 1 S 2 ) 1 2 (S 1 S 3 ) 1 2 (S 1 S 4 ) 1 2 (S 2 S 3 ) 1 i 2 (S 2 S 4 )+(S 3 S 4 ) E,2 1 2 h i (S 1 S 3 ) (S 1 S 4 ) (S 2 S 3 )+(S 2 S 4 ) Global E = 4 N " X tetra E,1 2 + X tetra E,2 2 # " X # Local E = 4 N 2 E,1 + 2 E,2 tetra 23

24 Global E = 4 N " # X 2 X 2 E,1 + E,2 tetra tetra " X # Local E = 4 N tetra 2 E,1 + 2 E,2 Local E = N T h h( Local E ) 2 i h Local E i 2i 24

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26 Neutron Structure Function f(q) =h S? (q) 2 i S? (q) =S S q/q 2 S(q) = X r i S i exp(iq r i ) 26

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

28 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

29 Thanks for your attention 29