Orbitals, reduced dimensionality and spin gaps and insulator-metal transitions

Orbitals, reduced dimensionality and spin gaps and insulator-metal transitions D.Khomskii Cologne University, Germany D.Kh. Physica Scripta (Comments Cond.Mat.Phys.) 72, CC8 (2005) D.Kh. Progr.Theor. Phys. Suppl.159, 319 (2005)

Introduction Directional nature of orbitals and exchange interactions Spin gaps on dimers; metal-insulator transitions Orbital-driven Peierls transitions Spin-gaps on trimers Charge disproportionation in Jahn-Teller systems Conclusions

Degrees of freedom charge Charge ordering (r) (monopole) Ferroelectricity P or D (dipole) Q (quadrupole) Spin Magnetic ordering Lattice Orbital ordering

Heisenberg Hamiltonian for a dimer Dimer: 1 2 H JS 1 S 2, J 0 Singlet (S=0): Тriplet (S=1): E S 3 4 J E T 1 4 J Ground state spin singlet

Goodenough-Kanamori-Anderson rules: 1. Exchange interaction of two half-filled orbitals is strong and antiferromagnetic (AFM) 2t U J AFM 2 2. Exchange interaction of half-filled and empty (or doubly-filled) orbitals is weak and ferromagnetic (FM) J FM t U ( U 2 2 J H J H )

Antiferro. orbital ordering in YTiO 3 gives ferromagnetism с

P Y R О X E N E S Diopside: CaMgSi2O6 Spodumene: LiAlSi2O6 Aegirine: NaFeSi2O6 Kosmochlore: NaCrSi2O6 Jade: NaAlSi2O6 Minerals

Crystal structure of pyroxenes Isolated chains of Me 3+ O 6 octahedra, sharing common edge, divided by (Si,Ge)O 4 tetrahedra

Spin gap in NaTiSi 2 O 6 300 K 100 K 8 H = 1T x 10 4 (emu/mol) 6 4 2 0 M. Isobe et al., J. Phys. Soc. Jpn. 71, 1423 (2002). 0 100 200 300 400 500 600 700 c Temperature (K) Bonner-Fisher curve for the S=1/2 Heisenberg linear AFM chain J/k B = 295 K Ti 3+ (3d 1 ;S=1/2) O 2- Natural explanation - formation of Ti-Ti singlet dimers!

NaTiSi 2 O 6 : calculations in LDA+U scheme 1 2 Ferro-orbital ordering AFM! - hopping -Coulomb repulsion

J intra U dd = 3.3 ev, J H = 0.8 ev J inter Energy gap: 1.8 ev Exchange in a dimer J intra = 396 K (AFM) Between dimers J inter = -5 K (FM) NaTiSi 2 O 6 chain consisting of dimers! Orbital ordering reduces dimensionality from 1-d to 0-d S. Streltsov, O. Popova, D. Khomskii PRL 96, 249701 (2006)

La 4 Ru 2 O 10 P.Khalifah et al, Science 297, 2237 (2002)

Hua Wu et al., PRL 96, 256402 (2006)

VO2: metal-insulator transition at 70 C V 4+ (t 1 2g) T>T c : all three orbitals equally populated, almost isotropic T<Tc:orbital repopulation; predominantly orbitals along the chains occupied. Peierls-like dimerization Switching of orbital occupation XAS: Haverkort et al., Phys. Rev. Lett. 95, 196404 (2005)

Spinels The B-site pyrochlore lattice: geometrically frustrated for AF

D.Kh. & T.Mizokawa, PRL 94, 156402 (2005)

M.Schmidt et al., PRL 92, 056402 (2004)

Experimental observations: Li 2 RuO 3 Miura et al JPSJ 07 Possible spin-singlet dimerized phase in 2D?!

t g2 orbitals on a honeycomb lattice: reduction to 0-dimensional case!

The ground state manifold Conclusion: The ground state manifold is generated by hard-core dimer coverings: Extensive orientational degeneracy Each spin is bound into spin-singlet Spin Gap

G.Jackeli and D.Kh., PRL 100, 147203 (2008)

Metallic bonding vs bond-charge repulsion and Jahn-Teller effect (LiVO2 vs NaVO2) Suppose I make orbital ordering with orbitals on two sites directed towards one another A question: will this bond become shorter or longer? a) Shorter: to enhance the covalency on this bond and to gain corresponding energy b) Longer: there is bond-charge repulsion of electronic clouds on these orbitals, and to reduce it one better moves these ions further apart

Same in TiI 2 (G.Meyer); in LiVS 2 (H.Takagi) H.Pen et al., PRL 78, 1323 (1997)

Orbital ordering in NaVO2 (T.M.McQueen, R.J.Cava et al, PRL 101, 166402 (2008)) Here ''red'' and ''blue'' occupied bonds become longer! (V-V ave =2.996Å, V-V short =2.977Å, V-V long =3.015Å - all bigger than R c Goodenough ~ 2.94Å)

Why such a difference? Li is smaller, the average V-V distance (2.84Å) is smaller, the system is closer to the crossover to itinerant behavior. The covalency wins, bonds become shorter. Na, and V-V distances are larger (3.00Å) - electrons are more localized, at larger distances orbital overlap decays rapidly and is less important, and bond-charge repulsion starts to dominate - bonds become longer. Another factor acting in the same direction in NaVO2 - local Jahn-Teller effect. Orbital occupation in NaVO2 correspond to local contraction of VO6 octahedra along the bond not containing occupied orbitals - and as a result the occupied bonds become longer. Local JT effect operates for localized electrons (NaVO2 ), and weakens and disappears closer to itinerant regime (LiVO2 )

E Average bond length r c less than critical: r c r covalency t(r) Minimum (equilibrium bond length) shifts to the left

E Average bond length r c more than critical: bond-charge repulsion r c r Minimum (equilibrium bond length) shifts to the right Thus the result depends on the proximity to localizeditinerant crossover, and gives opposite results in two regimes!

AlV2O4 Horibe et al., PRL 96, 086406 (2006)

Charge disproportionation close to Mott transition (in systems with Jahn-Teller ions) JT distortion for localized electrons. For itinerant electrons (metal; bands) no JT distortion T Param. M-H ins. without OO metal-ins. crossover metal OO (JT) ins.? t/u

Charge disproportionation in transition metal compounds Many compounds with (nominally) Fe 4+ 2Fe 4+ (t 2g3 e g1 ) Fe 3+ (t 2g3 e g2 ) + Fe 5+ (t 2g3 e g0 ) (CaFeO 3 ; Sr 3 Fe 2 O 7 ; Sr 2/3 La 1/3 FeO 3 ) Nickelates RNiO (Low Spin); AgNiO 3 2 (R.Coldea) 2Ni 3+ (t 2g6 e g1 ) Ni 2+ (t 2g6 e g2 ) + Ni 4+ (t 2g6 e g0 ) => in effect no orbital degeneracy left!

Experimental resistivity M. Abd-Elmeguid, R. Lenzdorf et al, Cologne

Charge disproportionation takes place at ambient P and low temperature. No JT distortion!

T Param. M-H ins. without OO metal-ins. crossover OO (JT) ins. I.Mazin, M.Abd-Elmeguid, D.Kh. et al, CO ins. metal CO metal t/u Phys. Rev. Lett. {\bf 98}, 176406 (2007).

Conclusions Orbital ordering largely determines magnetic properties of correlated oxides Directional nature of orbitals leads to a reduction of dimensionality (from 2-d/3-d to 1-d; from 2-d to 0-d) Often it leads to the formation of spin-gap states (on dimers; on trimers) Possible role in insulator-metal transitions Jan-Teller character important for charge disproportionation close to Mott transitions?