Modelling of Phase Transitions in R VO 3 Perovskites Andrzej M. Ole - - PowerPoint PPT Presentation

Hvar, 28 Sep 08 1

Modelling of Phase Transitions in RVO3 Perovskites

Andrzej M. Oleś

• M. Smoluchowski Institute of Physics, Jagellonian University, Cracow

Max-Planck-Institut für Festkörperforschung, Stuttgart Concepts in Electron Correlations Hvar, 28 September 2008

• Peter Horsch

Max-Planck-Institut FKF, Stuttgart

• Giniyat Khaliullin

Max-Planck-Institut FKF, Stuttgart

• Louis Felix Feiner

Philips Research Laboratories, Eindhoven

Hvar, 28 Sep 08 2

Outline

• Spin-orbital entanglement
• Phase diagram of RVO3: TOO and TN
• Spin-orbital model with orbital-lattice coupling
• Role of Jahn-Teller interaction and orbital-lattice term
• Variation of energy scales with ionic radius rR

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 3

Challenge: Phase Diagram of RMnO3

[J.-S. Zhou and J.B. Goodenough, PRL 96, 247202 (2006)]

LaMnO3 is a JT insulator

Changes for Lu <= La: TJT increases Magnetic interactions compete, TN decreases A-AF order => E-AF order

No microscopic model

Hvar, 28 Sep 08 4

Challenge for the Theory in the RVO3 Perovskites

[S. Miyasaka et al. PRB 68, 100406 (2003)]

TOO TN1

Microscopic model

Hvar, 28 Sep 08 5

Complementary behavior of spins and orbitals

Review of this field: Focus on Orbital Physics New Journal of Physics 2004-2005 http://www.njp.org LaVO3 t2g orbitals LaMnO3 eg orbitals C-AF A-AF Goodenough-Kanamori rules: AO order supports FM spin order FO order supports AF spin order

Are these rules sufficient?

AF phases with some FM bonds

Hvar, 28 Sep 08 6

Frustration of orbital interactions

) ( ) ( γ γ j ij i

• rb

T T J H

∑

> <

=

frustrated triangular lattice

• rder out of

disorder square lattice: no frustration Cubic symmetry of the orbital interactions: SU(2) symmetry for spins: Depend on bond direction => frustration: Different Ti components interact along each cubic direction γ=a,b,c 3D models at large U: (1) spins AF (Néel) order (2) orbitals

• rbital liquid

∑

> <

• =

ij j spin

S S J H

i

[L.F. Feiner and AMO, PRB 71, 144422 (2005)]

Hvar, 28 Sep 08 7

Orbital degrees of freedom in superexchange

In t2g systems (d1,d2) two active flavors, e.g. yz and zx along c axis – are described by quantum operators:

} , , {

z i y i x i i

T T T T = r

At finite η>0 the orbital operators contain:

z j z i y j y i x j x i j i

T T T T T T T T + − ≡ ⊗ r r

eg orbitals t2g orbitals . , ,

2 1 2 1 2 1 z i z i y i y i x i x i

T T T σ σ σ = = =

Orbital interactions have cubic symmetry

Orbital quantum numbers are not conserved ! Spin-orbital superexchange in RVO3 perovskites:

Hvar, 28 Sep 08 8

Spin-Orbital Model for RVO3 (R=La,Y, …)

t2g

2 configurations of V3+ ions with S =1 spins

each orbital is inactive along one axis

A.B. Harris et al., PRL 91, 087206 (2003)

t2g hopping For T<Ts xy orbitals are occupied:

Superexchange for t<<U (at JH=0):

Energies of t2g orbitals in RVO3

[G. Khaliullin et al., PRL 86, 3879 (2001)]

Hvar, 28 Sep 08 9

[AMO, GK, PH, and LFF, PRB 72, 214431 (2005)]

Superexchange: Multiplet structure in d-d excitations

Follows from three Racah parameters (Griffith, 1971): single parameter: η=JH /U high-spin low-spin

Hvar, 28 Sep 08 10

Superexchange and Optics in Cubic Vanadates

Virtual transitions across the Hubbard gap on bond <ij> determine magnetism. Same d-d transitions appear in optics. Strength of absorption into different multiplet states is linked to the magnetic

• rder (high-spin and low-spin states)

Partial sum rules follow from subdivision

• f full expression at finite JH :

η =JH/U, R =1/(1-3η), r =1/(1+2η) J=4t2/U

Hvar, 28 Sep 08 11

Superexchange in Mott insulators (t<<U)

Spin-orbital superexchange model at orbital degeneracy (γ=a,b,c - cubic axes) contains orbital operators

• f cubic symmetry

By averaging over orbital (dis)ordered state one finds effective spin model:

∑ ∑

⋅ + ⋅ =

c ab

ij ij j i ab j i c s

S S J S S J H

Here spin and orbital operators are disentangled

) ( ) ( γ γ ij ij

K and J

) (γ γ ij

J J ≡

( )

[ ]

• rb

j i j i j i j i j i

• rb

H K S S J J H ij H J H + + ⋅ = + =

∑∑ ∑∑

> < > < γ γ γ γ γ ) ( ) ( ) (

) ( r r

Spin AF Heisenberg model for one orbital (e.g. in high-Tc, t-J model, J=4t2/U):

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ∑∑

> <

4 1

j i j i

S S J H r r

γ

Spin interactions have SU(2) symmetry FM superexchange bonds are also possible (e.g. in A-AF and C-AF phases)

[AMO, GK, PH, and LFF, PRB 72, 214431 (2005)]

Hvar, 28 Sep 08 12

[A.M. Oleś et al., PRL 96, 147205 (2006)]

Spin-orbital entanglement in t2g models

d1 d2

In the shaded regions Jij is negative FM Sij is negative AF Tij is negative AO for d1 and d2 t2g models => GK rules are violated

) (γ ij ij

J J ≡

Definition of Jij is meaningless => entanglement

If Cij<0, spin and orbital operators are entangled

Sij – spin correlations Tij – orbital correlations × Cij – spin-orbital correlations

η=JH /U

S=1/2 S=1

Hvar, 28 Sep 08 13

Orbital fluctuations in C-AF phase of LaVO3

[G. Khaliullin et al., PRL 86, 3879 (2001)]

t2g

2 configuration of V3+ ions:

{a,b}={yz,zx} orbital fluctuations on the bonds || c axis ⇒ finite FM interaction -Jc>0 at η=0 (without Hund’s exchange!) ⇒ comparable values of AF Jab and FM -Jc at η=0.13 η=JH /U=0.13 Exchange constants in C-AF phase for increasing Hund´s exchange η shadow

Hvar, 28 Sep 08 14

Structural and Magnetic Transitions in RVO3

[S. Miyasaka et al. PRB 73, 224436 (2006)]

Characteristic features: (1) G-type OO and C-AF coexist; (2) two magnetic transitions (G-AF and C-AF phase) in YVO3; (3) G-type OO below TOO (4) C-AF order below TN1 Problem in the theory: Understanding the phase diagram

• f the RVO3 perovskites using the

microscopic spin-orbital model

G-AF phase C-AF phase

Hvar, 28 Sep 08 15

GdFeO3 –like Lattice Distortion in RVO3

[Eva Pavarini et al., New J. Phys. 7, 188 (2005)]

Distortions of VO6 are characterized by V-O-V bond angle and rotation angle with respect to c axis Lattice distortion:

Hvar, 28 Sep 08 16

Spin-Orbital-Lattice Coupling in RVO3

Model includes: (1) spin-orbital superexchange for S=1 spins and τ =1/2 pseudospins; (2) crystal field Ez induced by GdFeO3 distortions

• - it supports C-type OO with wavevector

(3) Jahn-Teller interaction Vab for the bonds in ab planes; (4) cooperative interaction ||c axis: TOO=TN1 at Vc=0.26J (in LaVO3); (5) orbital-lattice coupling term Hu Parameters: J, Ez, Vab and g (in Hu) Have to determine self-consistently singlet correlations η=JH /U=0.13 Hund’s exchange is fixed

Hvar, 28 Sep 08 17

Crystal Field Splitting and Orbital Interactions

Crystal field splitting increases with tilting angle

:

Orbital interaction (the JT term) follows the crystal field term: Both terms favor C-type OO in RVO3

[P. Horsch, AMO, G. Khaliullin, PRL 100, 167205 (2008)]

=> TOO increases with increasing distortion

Hvar, 28 Sep 08 18

Orbital-Lattice Interaction and Orbital Polarization

Interaction with the lattice favors orbital polarization in eigenstates: Distortion u contains the joint effect of the lattice u0 and Fast increase of the effective coupling with tilting is consistent with the rapid decrease of TOO for small values of ionic radius rR

Parameters of the spin-orbital model: Orbital-lattice interaction:

[P. Horsch, AMO, G. Khaliullin, PRL 100, 167205 (2008)]

Hvar, 28 Sep 08 19

Parameter changes due to increasing titling of VO6

Increasing tilting reduces the V-O-V bond angle

Microscopic parameters of the model (1) for varying V-O-V bond angle; parameters: Assuming that rotation angle we deduced that the CF varies Similar dependence for the JT term: Orbital-lattice parameter geff increases fast for decreasing V-O-V bond angle => TOO decreases

=> TOO increases

Hvar, 28 Sep 08 20

Cluster method for the <ij> bond along c axis

Order parameters determined self-consistently: <Sz> 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:

a b c zx yz C − AF / G − AO

• <Sz>
• <τ z>
• <Sz>

+<τ z>

TN1 is given by <Sz>=0 TOO is given by <τ z>=0

Below TN1, TOO: <Sz> and <τ z> are finite

Hvar, 28 Sep 08 21

Spin and orbital order in C-AF & G-OO Phase

Parameters: Superexchange and JT term induce G-OO below TOO:

Orbital polarization

increases from La to Sm and does not change at TOO Note: is finite for Spin and orbital order occur simultaneously in LaVO3 Spin and orbital order parameters

[P. Horsch, AMO, L.F. Feiner, G. Khaliullin, PRL 100, 167205 (2008)]

Hvar, 28 Sep 08 22

Calculated Phase Diagram of RVO3 (R =Lu,…,La)

Parameters: Important features: (1) Spin and orbital order couple to each other (2) TOO increases Y <= La due to large JT interaction (3) TOO decreases Lu <= Y due to large lattice term gu (4) TN1 decreases Lu <= La by spin-orbital coupling Ionic size varies with tilting: where

J = 200 K For LaVO3 TN1=147 K (exp: TN1=143 K)

[P. Horsch, AMO, L.F. Feiner, G. Khaliullin, PRL 100, 167205 (2008)]

Hvar, 28 Sep 08 23 23

Evolution of energy scales along RVO3 series

Orbital interactions in ab planes increase from La towards Lu due to increasing orbital-lattice (Jahn-Teller) Vab contribution Width of the magnon band decreases by a factor ~2 due to spin-orbital coupling (by 1.8 from LaVO3 to YVO3)

For geff=0 only weak decrease of WC-AF (dotted line, contradicts experiment)

[P. Horsch et al., PRL 100, 167205 (2008)]

Reduction of the magnon energy due to suppression of

Hvar, 28 Sep 08 24

Surprising correlation between geff, <τ x> and u

Orbital polarization <τ x>, coupling constant geff , distortion u0 at T=0, distortion u1 above TN1

• all increase Lu <= La

and follow each other ! Lattice strongly couples to {yz,zx} orbitals by geff and suppresses orbital

• rder <τz> and dynamics

Prediction: (1) Upper bound for g2/K (almost constant!); (2) A lower bound for K ~ 100 meV.

[P. Horsch, AMO, L.F. Feiner, G. Khaliullin, PRL 100, 167205 (2008)]

Hvar, 28 Sep 08 25

Towards orbital liquid in cubic vanadates ?

[J. Fujioka et al.., PRL 97, 196401 (2006)] [J. Fujioka et al.., PRB 72, 024460 (2005)]

La1-xSrxVO3

Insulator-metal transition at xc=0.17

No anisotropy for x>xc ⇒ melting of orbital order ⇒ t2g orbital liquid with C-AF order

Hvar, 28 Sep 08 26

Conclusions: Spin–orbital–lattice coupling

• 1. Spin-orbital superexchange model for RVO3 explains the

experimental data for the orbital transition at TOO and the magnetic transition at TN1 after including the crystal field, JT term and orbital-lattice coupling

• 2. Magnetic energy scale is reduced when ionic radius rR decreases
• 3. Orthorhombic lattice distortion u modifies orbital fluctuations

Future: second magnetic transition at TN2 in RVO3 compounds => role of the crystal-field splitting in switching from G-OO to C-OO

[preliminary: AMO, P. Horsch, and G. Khaliullin, PRB 75, 184434 (2007)]

Doped t2g systems: orbital liquid and its consequences

Hvar, 28 Sep 08 27

Thank Thank you you for for your

• ur kind

ind attention! ttention!

Hvar, 28 Sep 08 28

Absence of hole confinement for t2g orbitals

[M. Daghofer, K. Wohlfeld et al., PRL 100, 066403 (2008)]

VCA SCBA

Orbital t-Jz model

Hole propagates due to three-site terms:

Hvar, 28 Sep 08 29

Experimental Challenge for the Theory in RVO3

[S. Miyasaka et al. PRB 68, 100406 (2003)]

TOO TN1