Orbital Order in Titanium and Vanadium Spinels

1. “Entanglement in Spin & Orbital Systems” Krakow, 18-22 June 2008 Orbital Order in Titanium and Vanadium Spinels Sergio Di Matteo Équipe de Physique des Surfaces et Interfaces Institut de Physique de Rennes Université de Rennes 1 (France)

2. Main publications on the subject: S.Di Matteo, G.Jackeli, C.Lacroix, N.B.Perkins, Physical Review Letters93, (2004) 077208 Valence bond crystal in a pyrochlore antiferromagnet with orbital degeneracy. S. Di Matteo, G. Jackeli, N.B. Perkins, Physical Review B 72 (2005) 024431-1 à 15 Valence-bond crystal and lattice distortions in a pyrochlore antiferromagnet with orbital degeneracy. S. Di Matteo, G. Jackeli, N.B. Perkins, Physical Review B 72 (2005) R020408 Orbital order in vanadium spinels. In collaboration with: G. Jackeli, N.B. Perkins and C. Lacroix

3. Overview of the talk • General considerations The case of MgTi2O4 The case of MV2O4 (M=Zn, Mg, Cd) Conclusions

4. Frustrated antiferromagnets Spinels: AB2O4 (pyrochlore lattice of B-ions) - Highly frustrated antiferromagnet (eg, ZnCr2O4) ? ? Orbital order can suppress magnetic frustration and stabilize a well-defined order depending on the filling What does it happen when B-ions are orbitally degenerate ? (e.g., Ti3+ or V3+ ions)

5. From Isobe and Ueda (J. Phys. Soc. Jpn. 71, 1848 (2002)) The spinel t2g “series”  Insulator, Tt=260 K

6. energy energy eg eg 10 Dq 10 Dq t2g (S=1/2) t2g (S=1) Octahedral field and t2g orbitals (qualitatively different from eg orbitals…) MgTi2O4 Ti3+ 3d-electronic energy levels MV2O4 (M=Zn,Cd) V3+ 3d-electronic energy levels • Characteristics of t2g orbitals (vs eg): • larger degeneracy • they are less coupled to lattice • spin-orbit coupling can be important.

7. Experimental data for MgTi2O4 (I) Isobe and Ueda, Jour. Phys. Soc. Jpn. 71, 1848 (2002) • Metal-insulator transition at Tc=260 K • Cubic-to-tetragonal distortion • associated to the MIT 3) Magnetic susceptibility saturates at a value close to that of the spin-singlet VO2 and NaV2O5.

8. s l s l s Spin-singlet dimerization of short bonds ? Experimental data for MgTi2O4 (II) M. Schmidt et al., Phys. Rev. Lett. 92, 056402 (2004) 4) Neutron powder diffraction shows a tetragonal structure made of alternating short and long bonds forming a helix about tetragonal c-axis

9. tddd = 0.11 eV; tddp = 0.006 eV; For MgTi2O4, tdds = -0.32 eV; We can neglect tddp and tddd compared to tdds Theoretical description: the effective Kugel-Khomskii Hamiltonian 2nd order perturbation theory in t/U of the multi-orbital single-band Hubbard Hamiltonian : ;

10. in the plane ab (xy,xz,yz): Kugel-Khomskii Hamiltonian withdds-hopping (ddp and ddd neglected) Simplified orbital Hamiltonian : SE constants: J1  J(1+ h); J2  J(1 - h); J3  4J(1 - h); Relevant energy scales: J = t2 / U2  25 meV; and h = JH / U2  0.15

11. in the plane ab (xy,xz,yz): Kugel-Khomskii Hamiltonian withdds-hopping (ddp and ddd neglected) Simplified orbital Hamiltonian : SE constants: J1  J(1+ h); J2  J(1 - h); J3  4J(1 - h); Relevant energy scales: J = t2 / U2  25 meV; and h = JH / U2  0.15

12. y y y y z x x x x y x Hb0= J(1 - 2h)(4Si•Sj- 1) Hb2= 0 3) no orbitals of the same ‘kind’ as the plane (b2,b3) 2) one orbital of the same ‘kind’ as the plane (b1) Hb1= - J(1 + 2hSi•Sj + h/2) dyz dyz dxz dxy Hb3= 0 dyz dxy dyz dxy Bonds classification: 1) two orbitals of the same ‘kind’ as the plane (b0)

13. From Pi,xy + Pi,xz + Pi,yz = 1 2Nb0+Nb1= 4 ! h0: Eb0= -4J Eb1= -J Eb2= 0 Eb3= 0 Transformation of a dynamical problem into a combinatorial one Hb1= - J(1+2hSi•Sj+h/2) ; Hb2 = Hb3= 0 Hb0= J(1-2h)(4Si•Sj- 1) ;

14. All-A phase Covering of the unit cell: correlation of tetrahedra A-phase (Heisenberg 1D) : E = -2.77 J (1-2h) B-phase (AFM dimers): E = - (3 - 7/2 h) J C-phase (FM) : E = -2 J (1 +h) Mixed AC-phase: degenerate with B-phase ! (1 b0 and 2 b1 bonds in average) B-dimer phase

15. ‘True’ phase All B-phases are characterized by an equal number of b2 and b3 bonds in average: nb0=1; nb1=2; nb2=2; nb3=1 Mixed-B12 phase Covering of the unit cell: correlation of bonds Mixed-AC phase

16. There are situations where a SE energy gain at shorter bonds is present and it is not compensated by any SE loose at longer bonds… Triplet distortion of the tetrahedron (qualitatively)

17. with: Magnetoelastic energy minimization (quantitatively)

18. B3-phase Corresponding to a long-range helical dimerization of the spin pattern, with spin-singlets (dimers) located at short bonds, in a phase that is a (non-resonating) valence-bond crystal MgTi2O4 ground state is:

19. Experimental data for ZnV2O4 S.-H. Lee et al., Phys. Rev. Lett. 93, 156407 (2004) M. Reehuis et al., Eur. Phys. Jour. B 35, 311 (2003) M. Onoda & J. Hasegawa, JPCM 15, L95 (2003) • Two very close transition temperatures: a structural one (cubic-to-tetragonal) at Tc=52 K and a (3D) AFM one at TN=44 K (and similar for CdV2O4 and MgV2O4) • 2) An AFM (Néel) 1D order, along the directions [110], [110] with a saturated moment m=0.6mB, directed along the [001] direction between the two transition temperatures. • 3) A tetragonal space group (I41/amd) apparently at odds with the marked anisotropy in the structure factor S(Q)…

20. Tsunetsugu & Motome’s ground state PRB 68, 060405R (2003) Tchernyshyov’s objection PRL 93, 157206 (2004) 1D AFM chains (but JT not predetermined and incorrect space group) Correct space group, but unable to predict the sign of JT distortion

21. V3+ 3d-electronic energy levels Equivalence of t2g with p electrons in Oh field with L’~ -L 1S → 1-fold 1D → 5-fold Extra degeneracy-lift due to spin-orbit… 3d2 3PJ→ 9-fold J = 0 multiplets energy J = 1 G3(2) eg 2l spin-orbit J = 2 10 Dq Oh-field G5(3) t2g (S=1) Superexchange gain is of the same order of magnitude, if not bigger than SO… The physics of V3+ ion (also, e.g., V2O3, LuVO3, …)

22. The case of CdV2O4: the effective SE+SO Hamiltonian (only dds-hoppings: ddp and ddd neglected) with: (for holes !) With J1 = J h/(1-3h); J2 = J (1-h)/(1-3h); J3 = J (1+h)/(1+2h); and h=JH/U1  0.10 ; J=t2/U1  10 meV ; l > 0 (e.g., 13 meV ) L’ acts in an effective L = 1 subspace spanned by t2g states  fundamental the relative ratios l/J & h !

24. The case of CdV2O4: the effective SE+SO Hamiltonian (only dds-hoppings: ddp and ddd neglected) with: (for holes !) With J1 = J h/(1-3h); J2 = J (1-h)/(1-3h); J3 = J (1+h)/(1+2h); and h=JH/U1  0.10 ; J=t2/U1  10 meV ; l > 0 (e.g., 13 meV ) L’ acts in an effective L = 1 subspace spanned by t2g states  fundamental the relative ratios l/J & h !

25. = b3 = b1 Bond classification and energies Hb1= - J(1+2h+hSi•Sj) ; Hb2 = Hb3= - J(1-h)(1 - Si•Sj) Hb0= 0 ; = mixed (complex) C = dyz = dxz E= -2J1-2J2 E= -J0-2J1-2J2 E= -J0-2J1-2J2 -l/4 = dxy C = (dxz +i dyz)/2 = (dxz-i dyz)/2 E= -J1-3J2 -l/2 E= -J1-5J2/2-l

26. Interchain coupling: 1) for COO phase : J’ = (J2-2J0) / 4 ~ l = l / J 2) for Tsun&Mot: J’’ = Jh/(1-3h) For h ~ 0.1  J’ = 0.15 J & J’’= 0.14 J Phase diagram and ground state

27. Conclusions: 1) The different behavior of MgTi2O4 (spin-dimers, ROO) and of CdV2O4 (1D Heisenberg AFM, COO) can be ascribed to the different filling (21 !). 2) Particularly for MgTi2O4, where we could solve our electronic model exactly, the agreement with the experimental data is satisfying. • In spite of these results, I have a negative remark: the theory is still too much at the level of a “art” (no easy protocol to follow) and the physical intuition is still too dependent on known experimental results (no strict predictability).