Modelling of Phase Transitions in R VO 3 Perovskites Andrzej M. Ole ś M. Smoluchowski Institute of Physics, Jagellonian University, Cracow Max-Planck-Institut für Festkörperforschung, Stuttgart Concepts in Electron Correlations Hvar, 28 September 2008 • Max-Planck-Institut FKF, Stuttgart Peter Horsch • Max-Planck-Institut FKF, Stuttgart Giniyat Khaliullin • Philips Research Laboratories, Eindhoven Louis Felix Feiner Hvar, 28 Sep 08 1
Outline • Spin-orbital entanglement • Phase diagram of R VO 3 : T OO and T N • Spin-orbital model with orbital-lattice coupling • Role of Jahn-Teller interaction and orbital-lattice term • Variation of energy scales with ionic radius r R This research was supported by the Foundation for Polish Science (FNP) and by the Polish Ministry of Science and Education under Project No. N202 068 32 / 1481 Hvar, 28 Sep 08 2
Challenge: Phase Diagram of R MnO 3 LaMnO 3 is a JT insulator Changes for Lu <= La: T JT increases Magnetic interactions compete, T N decreases A -AF order => E -AF order No microscopic model Hvar, 28 Sep 08 3 [J.-S. Zhou and J.B. Goodenough, PRL 96 , 247202 (2006)]
Challenge for the Theory in the R VO 3 Perovskites T OO T N1 Microscopic model Hvar, 28 Sep 08 [S. Miyasaka et al . PRB 68 , 100406 (2003)] 4
Complementary behavior of spins and orbitals AF phases with some FM bonds Goodenough-Kanamori rules : C -AF A -AF AO order supports FM spin order FO order supports AF spin order Are these rules sufficient? Review of this field: Focus on Orbital Physics New Journal of Physics LaVO 3 LaMnO 3 2004-2005 t 2g orbitals e g orbitals http://www.njp.org Hvar, 28 Sep 08 5
Frustration of orbital interactions ∑ = • SU(2) symmetry for spins: H J S S spin j i < > ij frustrated order out of square lattice: triangular lattice disorder no frustration ∑ γ γ = Cubic symmetry of the orbital interactions: ( ) ( ) H J T T orb i j < > Depend on bond direction => frustration : ij Different T i components interact along each cubic direction γ= a,b,c 3D models at large U : (1) spins AF (Néel) order (2) orbitals orbital liquid Hvar, 28 Sep 08 6 [L.F. Feiner and AMO, PRB 71 , 144422 (2005)]
Orbital degrees of freedom in superexchange In t 2g systems ( d 1 , d 2 ) two active flavors, e.g. yz and zx along c axis – are described by quantum operators: r T = x y z { T , T , T } i i i i e g orbitals t 2g orbitals y y x = σ x = σ z = σ z 1 1 1 T , T , T . i i i i i i 2 2 2 At finite η > 0 the orbital operators contain: Orbital interactions r r ⊗ ≡ − + x x y y z z have cubic symmetry T T T T T T T T i j i j i j i j Orbital quantum numbers are not conserved ! Spin-orbital superexchange in R VO 3 perovskites: Hvar, 28 Sep 08 7
Spin-Orbital Model for R VO 3 ( R =La,Y, …) 2 configurations of V 3+ ions with S =1 spins t 2g t 2g hopping each orbital is inactive along one axis For T<T s xy orbitals are occupied: A.B. Harris et al ., PRL 91 , 087206 (2003) Energies of t 2g orbitals in R VO 3 Superexchange for t<<U (at J H =0): Hvar, 28 Sep 08 8 [G. Khaliullin et al ., PRL 86 , 3879 (2001)]
Superexchange: Multiplet structure in d-d excitations Follows from three Racah parameters (Griffith, 1971): low-spin high-spin single parameter: η =J H /U Hvar, 28 Sep 08 9 [AMO, GK, PH, and LFF, PRB 72 , 214431 (2005)]
Superexchange and Optics in Cubic Vanadates Partial sum rules follow from subdivision of full expression at finite J H : � Virtual transitions across the Hubbard gap on bond <ij> determine magnetism. � Same d-d transitions appear in optics . � Strength of absorption into different η = J H /U , R =1/(1-3 η ), r =1/(1+2 η ) multiplet states is linked to the magnetic order (high-spin and low-spin states) J=4t 2 /U Hvar, 28 Sep 08 10
Superexchange in Mott insulators ( t<<U ) Spin AF Heisenberg model for one orbital (e.g. in high- T c , t-J model, J=4t 2 /U ): r r = ∑∑ ⎛ ⎞ 1 − ⋅ ⎜ ⎟ J Spin interactions have SU(2) symmetry H S S i j ⎝ ⎠ 4 γ < > i j Spin-orbital superexchange model at orbital degeneracy ( γ =a,b,c - cubic axes) [ ] ( ) r r ∑∑ ∑∑ γ + = = + ( ) γ ⋅ + γ ( ) ( ) J H ( ij ) H J H H J S S K orb orb i j i j i j γ < > γ < > i j i j γ γ ( ) ( ) contains orbital operators of cubic symmetry J and K ij ij By averaging over orbital (dis)ordered state one finds effective spin model : ∑ ∑ = ⋅ + ⋅ ( γ H J S S J S S ) ≡ s c i j ab i j J J γ ij ij ij c ab FM superexchange bonds are also possible (e.g. in A -AF and C -AF phases) Here spin and orbital operators are disentangled Hvar, 28 Sep 08 11 [AMO, GK, PH, and LFF, PRB 72 , 214431 (2005)]
Spin-orbital entanglement in t 2g models If C ij <0, spin and orbital operators are entangled d 1 S ij – spin correlations S=1/2 T ij – orbital correlations × C ij – spin-orbital correlations ( γ ) ≡ d 2 J J ij ij S=1 In the shaded regions J ij is negative FM S ij is negative AF T ij is negative η =J H /U AO for d 1 and d 2 t 2g models Definition of J ij is meaningless => GK rules are violated => entanglement Hvar, 28 Sep 08 12 [A.M. Ole ś et al ., PRL 96 , 147205 (2006)]
Orbital fluctuations in C -AF phase of LaVO 3 2 configuration of V 3+ ions: t 2g { a,b } = { yz,zx } orbital fluctuations on the bonds || c axis ⇒ finite FM interaction - J c >0 at η =0 ( without Hund’s exchange! ) ⇒ comparable values of AF J ab and FM - J c at η =0.13 Exchange constants in C -AF phase for shadow increasing Hund´s exchange η η =J H /U= 0.13 Hvar, 28 Sep 08 [G. Khaliullin et al ., PRL 86 , 3879 (2001)] 13
Structural and Magnetic Transitions in R VO 3 Characteristic features: (1) G -type OO and C -AF coexist; (2) two magnetic transitions ( G -AF and C -AF phase) in YVO 3 ; (3) G -type OO below T OO (4) C -AF order below T N1 [S. Miyasaka et al . PRB 73 , 224436 (2006)] Problem in the theory: Understanding the phase diagram of the R VO 3 perovskites using the microscopic spin-orbital model Hvar, 28 Sep 08 14 G -AF phase C -AF phase
GdFeO 3 –like Lattice Distortion in R VO 3 Distortions of VO 6 are characterized by V-O-V bond angle and rotation angle with respect to c axis Lattice distortion: Hvar, 28 Sep 08 15 [Eva Pavarini et al ., New J. Phys. 7 , 188 (2005)]
Spin-Orbital-Lattice Coupling in R VO 3 Model includes: (1) spin-orbital superexchange for S =1 spins and τ =1/2 pseudospins; (2) crystal field E z induced by GdFeO 3 distortions -- it supports C -type OO with wavevector (3) Jahn-Teller interaction V ab for the bonds in ab planes; (4) cooperative interaction || c axis: T OO = T N1 at V c =0.26 J (in LaVO 3 ); (5) orbital-lattice coupling term H u η =J H /U= 0.13 Hund’s exchange is fixed Have to determine self-consistently singlet correlations Hvar, 28 Sep 08 16 Parameters: J, E z , V ab and g (in H u )
Crystal Field Splitting and Orbital Interactions Both terms favor C -type OO in R VO 3 Crystal field splitting increases with tilting angle : Orbital interaction (the JT term) follows the crystal field term: => T OO increases with increasing distortion [P. Horsch, AMO, G. Khaliullin, PRL 100 , 167205 (2008)] Hvar, 28 Sep 08 17
Orbital-Lattice Interaction and Orbital Polarization Orbital-lattice interaction: Interaction with the lattice favors orbital polarization in eigenstates: Distortion u contains the joint effect of the lattice u 0 and Fast increase of the effective coupling with tilting is consistent with the rapid decrease of T OO for small values of ionic radius r R Parameters of the spin-orbital model: [P. Horsch, AMO, G. Khaliullin, PRL 100 , 167205 (2008)] Hvar, 28 Sep 08 18
Parameter changes due to increasing titling of VO 6 Increasing tilting reduces the V-O-V bond angle Assuming that rotation angle we deduced that the CF varies Similar dependence for the JT term: => T OO increases Microscopic parameters of the model (1) for varying V-O-V bond angle; parameters: Orbital-lattice parameter g eff increases fast for decreasing V-O-V bond angle => T OO decreases Hvar, 28 Sep 08 19
Cluster method for the < ij > bond along c axis Order parameters determined self-consistently: < S z > and < τ z >, with A bond < ij >|| c axis with MF terms due to its neighbours is solved Singlet correlations are renormalized to the exact result for the 1D chain: - < S z > - < τ z > zx Below T N1 , T OO : < S z > and < τ z > are finite - < S z > c +< τ z > T N1 is given by < S z >=0 yz b T OO is given by < τ z >=0 a C − AF / G − AO Hvar, 28 Sep 08 20
Spin and orbital order in C -AF & G -OO Phase Superexchange and JT term induce G -OO below T OO : Orbital polarization increases from La to Sm and does not change at T OO Note: Spin and orbital order parameters is finite for Parameters: Spin and orbital order occur simultaneously in LaVO 3 Hvar, 28 Sep 08 21 [P. Horsch, AMO, L.F. Feiner, G. Khaliullin, PRL 100 , 167205 (2008)]
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