1. Magnetic Interactions and Order-out-of-disorder in Insulating Oxides Ora Entin-Wohlman, A. Brooks Harris, Taner Yildirim Robert J. Birgeneau, Marc A. Kastner, Koichi Katsumata R. Ramirez, C. Broholm, J. W. Lynn TAU, BGU, U Penn, NIST, MIT, RIKEN, Lucent, JHU Les Houches summer school on Quantum Magnetism, June 2006 Amnon Aharony

2. 2 Lecture 3: Vanadates: Competing nn and nnn interactions yield Incommensurate order Competing anisotropies yield complex field dependent phase diagrams Ni and Co have very different magnetic structures Theoretical tools introduced in pervious lectures suffice to explain most features

3. 3 General outline: Cuprates Vanadates Lecture 3

4. 4 Buckled Kagome S=1 S=3/2

5. 5 Buckled Kagome

6. 6 Crystal Structure of Ni 3 V 2 O 8 c a b Cross-tieSpine Only magnetic (S=1) Ni ions are shown Cross-tie is FRUSTRATED ?

7. 7

8. 8 0 5 0 5 0 5 Magnetic Field (T) 0 2 4 6 8 10 Temperature (K) H || a H || b H || c

9. 9

10. 10 Specific heat Neutron scattering intensities in C, LTI and HTI Incommensurate wave vector Weak ferromagnetism in C phase

11. 11 CAF’ = Incommensurate? Paramagnetic HTI = High Temperature Incommensurate Phase LTI = Low Temperature Incommensurate Phase CAF = Antiferromagnetic + weakly ferromagnetic MAGNETIC PHASE DIAGRAM OF Ni 3 V 2 O 8

12. 12 MAGNETIC PHASE DIAGRAM OF Ni 3 V 2 O 8

13. 13 Theory Step I: Main interactions along spines: Superexchange, Ni — O — Ni and Ni — O — O--Ni O Ni OO Explain HTI, LTI, CAF

14. 14 Incommensurability? -- simplest model: HTI LTI At low T, anisotropy wins again  CAF (q locked in)

15. 15 Step II: Anisotropy comes from spin orbit interactions Spin-orbit interaction generates Antiferromagnetic bond-dependent spin anisotropy Also Dzyaloshinskii-Moria antisymmetric exchange Ni O Oxygen tilted along z D along y, AFM along x FM along z Bilinear coupling between staggered Moment along a and ferromagnetic Moment along c

16. 16 xy I1 I2 II xy I1 I2 II Step III: spin on cross-tie NI? Pseudodipolar interactions

17. 17 More recent results: Multiferroic behavior Ferroelectric moment along b, only in LTI phase! Can switch ferroelectric moment with magnetic field!

18. 18 PHASE DIAGRAM SPONTANEOUS POLARIZATION Magnetic Field (T) 0 5 0 5 0 5 0 2 4 6 8 10 Temperature (K) a b P(  C/m 2 ) T=5K P || b H || c T=4K H || a H || c 0 0.5 1 1.5 2 2.5 3 3.5 Magnetic Field (T)

19. 19 LANDAU THEORY WITH TWO ORDER PARAMETERS THIS DOES NOT WORK!! WE DO NOT BELIEVE IN ACCIDENTAL DEGENERACY (T P = T M ). ALSO BOTH M AND P DEPEND STRONGLY ON H, SO THEN, WHEN WE MINIMIZE WITH RESPECT TO P, P APPEARS ONLY WHEN M IS NONZERO :

20. 20 MAGNETOELECTRIC COUPLING where x,y are LTI or HTI and  = x,y,z In the HTI phase we have a single order parameter which has a node at some lattice site. About this site there will be inversion symmetry. So I  (q) =  (-q) =  (q)* I = inversion operator  q) =  (-q)* is an order parameter = 0 ( IH = H )

21. 21 MANETOELECTRIC INTERACTION Thus the trilinear magnetoelectric interaction is of the form H =  HTI  LTI P + d P 2 So, after we minimize with respect to P: P = const  HTI  LTI = const  LTI This qualitatively explains the dependence of P on T and H

22. 22 Confirm mean field trilinear term from microscopic Hamiltonian Can arise from DM and PD interactions

23. 23 B 2u -phonons Mode Number: 64 Mode Energy: 69.24 meV (experimental value is about 80 meV!) Mode Description: Two oxygen atoms connected to cross-tie Ni moves along b-axis, significantly effecting the Ni-O-Ni bond angle for the spine spins (see the animations; side and top views). Dipole Moment: 0.4612 (One of the largest dipole moment!) b-axis Spine-spins (a-axis) Cross-tie Connected to V

24. 24

25. 25 Phys. Rev. B, in press

26. 26 (Spins along spine parallel to each other)

27. 27 FM, d =0 AFM, d =1/2

28. 28 Theory x(J 3 )

29. 29 Quartic terms Higher harmonics Lock-in

30. 30 Dielecric constant Ferroelectricity???

31. 31 Conclusions: Vanadates are almost frustrated; interesting phase diagrams Can explain incommensurate phases by competing interactions Multiferroics!

32. 32 THE END