Universidad de Cantabria

Spain

P. García-Fernández*, J. A. Aramburu, M. T.Barriuso and M. Moreno

Origin of octahedral tilting in the

perovskites

XX International Symposium on the Jahn-Teller Effect

Fribourg, 16th - 20th August 2010

*email: garciapa@unican.es

garciapa@unican.es Origin of octahedral tilting in the perovskites 1/16

IntroductionPerovskite crystals present a large variety of properties

Electrical: Conducting, insulators, superconductorsMagnetic: Ferro, antiferro, diamagneticDielectric: Ferroelectricity, …

Many properties are sensitive to external influenceTemperature, pressure, strain, applied fields…

Changes associated to structural phase transitions!

Is tilting unrelated to the Pseudo Jahn-Teller?

Three main types of distortion

- Jahn-Teller effect in BX6 octahedra- Ferroelectric distortions- Tilting of the BX6 octahedra

electronicelectronic

steric

The simple perovskite ABX3 structure is cubic Pm3m

P.W. Woodward, Acta. Cryst. B53, 32 (1997)

IntroductionPerovskite structure is formed of alternating AX and BX2 planes

Tilting is associated with the rotation of BX6 octahedra

AB

X (F-, O2-) Two triply degenerate modes associated to them

AX

BX2

AX

In-phase

M3+ R3

+

garciapa@unican.es Origin of octahedral tilting in the perovskites 2/16

Opposite-phase

garciapa@unican.es Origin of octahedral tilting in the perovskites

Steric modelsTiltings appear due to the misfit size of the A cation

Goldschmidt ratio

How do the atoms fit inside the perfect cubic lattice if they are considered hard spheres?

BX2 plane AX plane

a=2R(B)+2R(X) a= R(A)+ R(X)2√2

2√2

R(B)+R(X)√2R(A)+R(X)τ = 1

τ ~1 cubicτ <1 tilting

τ >1 Ferroelectric (B site)

D.I. Brown, Acta Cryst. B 48, 553 (1992)

• The origin of ferroelectricity is not steric

• Ionic radii are empiric parameters depending on:• Why does A not move off-center when small?

- Ion charge- Ion coordination- Magnetic state

ReasonableBorn-Mayer

H.D. Megaw, Acta Cryst. B 24, 149 (1968)

3/16

garciapa@unican.es Origin of octahedral tilting in the perovskites

Steric models

Goldschmidt ratio

R(B)+R(X)√2R(A)+R(X)τ = 1

τ ~1 cubicτ <1 tilting

τ >1 Ferroelectric (B site)

D.I. Brown, Acta Cryst. B 48, 553 (1992)

• The origin of ferroelectricity is not steric

• Ionic radii are empiric parameters depending on:• Why does A not move off-center when small?

- Ion charge- Ion coordination- Magnetic state

?

ReasonableBorn-MayerCovalency!

H.D. Megaw, Acta Cryst. B 24, 149 (1968)

3/16

high-spin

t2g

eg

low-spin

antibonding σ-orbitals

antibonding π-orbitals

longer distances

garciapa@unican.es Origin of octahedral tilting in the perovskites 4/16

Steric modelsHow does it perform?

P.W. Woodward, Acta. Cryst. B53, 44 (1997)

R(B)+R(X)√2R(A)+R(X)τ = 1 τ ~1 cubic

τ <1 tilting0 tilt 1 tilt

LaAlO3LaCuO3LaCoO3LaNiO3

CaMnO3

SrRuO3

CaTiO3

SrZrO3

KAgF3KMgF3 1.032BaNbO3 1.031KZnF3 1.021KNiF3 1.014

KMnF3 0.978SrTiO3 1.009

1.0171.0141.0111.003

1.0121.0010.9730.9530.930

3 tilt

It usually works well enough to predict tendencies

It has problems in a system-by-system basis

How are tiltings connected to the crystal’s electronic properties?

Reasonable due to the small energy scale (100-500K) typical of these phase transitions

More advanced theories (bond-valence, etc.) with same backgroundare suppossed to overcome some of these problems

D.I. Brown, Acta Cryst. B 48, 553 (1992)

garciapa@unican.es Origin of octahedral tilting in the perovskites 5/16

Magnetic propertiesThe magnetic phase is dependent on the tiltings

Antiferromagnetic phases are common for no or low tiltingsFerromagnetic phases are common for large rotations

Consider: Superexchange interaction

P. Jeffrey Hay et al., J. Am. Chem. Soc. 97, 4884 (1975)

no tiltingstrong σ-bonding!

large tiltingweak σ-bonding!

metal (3d)

ligand (2p)

antiferromagnetic

metal (3d)

ligand (2p)

ferromagnetic

d-orbital energy is sensitive to distortion and covalency

Strong similarity to Pseudo Jahn-Teller?

garciapa@unican.es Origin of octahedral tilting in the perovskites 6/16

Local view of the distortionLocally, tilting is similar to bending in a linear molecule

Bending in molecules is related to the presence of empty d(xy,xz,yz)-orbitals on the metal

Is tilting associated to d(xy,xz,yz) orbitals?

CaF2

SrF2

BaF2

MgF2

bentbentbent

linear

Similarly KMgF3 does not present tilting, KCaF3 presents tilting

Z

X

KCaF3CaF2

P.Garcia-Fernandez et al., J. Phys. Chem. A, 111, 10409 (2007)

garciapa@unican.es Origin of octahedral tilting in the perovskites 7/16

MethodWe have studied the presence of tilting in the ionic series:

KMF3 M=Ca, Sc, …,Znwhere the population of d-orbitals increases along the row

d0 d10

What is the effect of population on t2g(xy,xz,yz) and eg(3z2-r2,x2-y2)?

low-spin high-spint2g

eg

garciapa@unican.es Origin of octahedral tilting in the perovskites 8/16

Computational detailsWe have performed full geometry optimization of the cell following R3z

+ and M3z+ modes

We have employed Hartree-Fock and Density-Functional-Theory- LDA- B3LYP (Hybrid Functional)

All calculations carried out with CRYSTAL06 package

Exact-exchange is necessary to correctly simulate magnetic states

k-space sampling: 8x8x8All electron localized gaussian basis set

- DZP qualityTight convergence parameters

M3+ R3

+

P4mbm I4mcm

12

8

4

00 1 2 3 4 5 6

Number of t2g electrons

θ(degrees)

Ti2+

Sc2+

Ca2+

Zn2+Cu2+Ni2+

V2+

garciapa@unican.es Origin of octahedral tilting in the perovskites 9/16

ResultsWith the previous method accurate results can be obtained

Cr2+

Mn2+

Fe2+

Co2+

Mn2+Cr2+

Fe2+Co2+

• Lattice parameters within 1% of experimental results• Angles within 1º of experimental results• Correct magnetic states for the lattice (AFM)

Is the population of the t2g orbital relevant to tilting?

High spin configuration

Larger rotation →

less t2g electrons

low-spin high-spin

t2g

eg

P. Garcia-Fernandez et al., J. Phys. Chem. Lett. 1, 647 (2010)

garciapa@unican.es Origin of octahedral tilting in the perovskites 10/16

KNiF3

KZnF3

KMnF3

400

300

200

100

0

‐1000 2 4 6 8 10 12

Energy (cm

‐1)

θ (degree)

Energy cross-section It is interesting to study the energy cross-section for different cases

(t2g3)

KZnF3 is almost unstable!

Comparing Ni2+ and Zn2+ radii

We would expect KZnF3 to be even more stable than KNiF3

(t2g6)

R(Ni2+)83pm

KZnF3 tends to tilting due to low-lying Zn 4s-orbitals → extra PJT contributionsDifferent crystals have different covalent bonding patterns and different tilting tendency

(t2g6)

R(Zn2+)74pm>

garciapa@unican.es Origin of octahedral tilting in the perovskites 11/16

CovalencyLet us visualize the rotation in a BX2 plane

garciapa@unican.es Origin of octahedral tilting in the perovskites 11/16

CovalencyLet us visualize the rotation in a BX2 plane

Tilting breaks σ-bondingThe overlap variation is

quadratic

garciapa@unican.es Origin of octahedral tilting in the perovskites 11/16

CovalencyLet us visualize the rotation in a BX2 plane

Tilting allows new overlapsThe effect is linear

Tilting breaks σ-bondingThe overlap variation

is quadratic

Tilting opens up new bonding channels:

garciapa@unican.es Origin of octahedral tilting in the perovskites 12/16

0.5

high‐spin low‐spin

0.5

0.5

CovalencyPseudo Jahn-Teller is associated with new covalencies

Where is charge concentrating? Compare KMnF3 cases

Density difference diagram with respect to spheric ionselectron concentration electron defect

High-spin system shows increased π-covalency

garciapa@unican.es Origin of octahedral tilting in the perovskites 13/16

CovalencySteric interactions → lower electron-electron repulsionPJT → new covalencies→ higher electron-nuclei interaction

Energy

(eV)

200

100

50

0

‐50

‐1000 2 4 6 8 10 12θ (degree)

150 Ven

VnnVee

Energy

(eV)

0.4

0.3

0.2

0.1

0

‐0.10 2 4 6 8 10 12

θ (degree)

Ven

Vee

Trends seem to agree with steric interpretation but:It is impossible to distinguish high and low spin KMnF3

Energy scale is much higher than that of the transition (55K)

high-spin KMnF3 vs low-spin KMnF3Total energy variation Variation difference (high-low)

High spin shows increased covalency with respect to low spin

garciapa@unican.es Origin of octahedral tilting in the perovskites 14/16

Are there no steric effects in the perovskites?The force constant of a system can be written:

K=K0+Kv = K0 -F2

Δ

vibronic termnegative (favors distortion)

describes the change in electron density

elastic termpositive (against distortion)

steric termssmaller A ions may favor distortion but

never produce it

0.5 0.5

NaMgF3 KMgF3

NaMgF3 shows extra covalency vs KMgF3

but covalency should be considered the main causePJT theorem

(Bersuker, 1980)

garciapa@unican.es Origin of octahedral tilting in the perovskites 15/16

OxidesLet us consider the spin-lattice coupling

CaMnO3

Steric models are clearly incorrect to describe this behavior

However, the effect is dependent on the fine details of the electronic structure and can only be explained by PJT

In some systems the vibrational frequencies associated to different magnetic states are different

Calculation of the tilting frequencies for FM and AFM states:

SrMnO3KMnF3

antiferromagnetic

ferromagnetic

193i1022i250i202i39i

J.H. Lee et al., Phys. Rev. Lett. 104, 207204 (2010)P. Garcia-Fernandez et al. sent to Phys. Rev. Lett.

garciapa@unican.es Origin of octahedral tilting in the perovskites 16/16

Conclusions• Tiltings have traditionally been attributed to steric effects, however, this approach contains many problems.

Thank you for your attention!

P. Garcia-Fernandez et al., J. Phys. Chem. Lett. 1, 647 (2010)

• Cannot explain tiltings in a system-by-system basis• Leads to unphysical results in WO3• Clearly unable to explain spin-lattice coupling

• The Pseudo Jahn-Teller model solves many of these inconsistencies• Allows to establish bridges between electronic properties and structure• When the B-ion is a transition metal the population of the t2g orbitals is of great importance in the rotation.

P. Garcia-Fernandez et al., sent to Phys. Rev. Lett.

garciapa@unican.es Origin of octahedral tilting in the perovskites 17/21

~xz; yz

~x2-y2~x2-y2

~3z2-r2~3z2-r2

~xy

eg

t2g

cubicperovskite

3d

Free ion tiltedperovskite

t1g

Ligand 2p

CovalencyFrom a local point the main effects of tilting on electronic structure can be depicted in an orbital diagram:

Does the change of electronicstructure influence any property?

E. Pavarini et al., New J. Phys. 7, 188 (2005)

Unusual localization of electronsin d1 transition metal containingperovskitesPJT increases gap which isassociated to localization

garciapa@unican.es Origin of octahedral tilting in the perovskites 19/21

OxidesLet us study an special system:

Its structure is that of a perovskitewith its A ion removed

In steric models the energy is gained by getting the anions closer to the A cation

However, in WO3 rigid rotations only increase the energy that way!

WO3

Minimum tolerance factor possible!

Large tiltings!

Does it make sense?

The energy gain must come from covalency!

RepulsionAttraction

garciapa@unican.es Origin of octahedral tilting in the perovskites 18/21

OxidesLet us compare two typical oxides:

BaTiO3

CaTiO3

τ

1.070.97

No tiltingTilting

What happens when we apply pressure?BaTiO3 does not rotate, as expected by large tolerance factor

Pressure

Increases tilting

Tilting!

Is this general?

KMgF3 1.03 No tilting No tilting

Some authors explain these effects by ion site compressibilityusing more empirical parameters!

The overlap due to distortion increases fast with pressure!

small PJT constant large PJT constant

Introduction

Perovskite structure is formed of alternating AX and BX2 planes

Tilting is associated with the rotation of BX6 octahedra

AB

X (F-, O2-)

Two triply degenerate modes associated to them

Introduction

Perovskite structure is formed of alternating AX and BX2 planes

Tilting is associated with the rotation of BX6 octahedra

AB

X (F-, O2-)

Two triply degenerate modes associated to them

In-phase Phase-opposition

M3+ R3

+