02/04/14

1

Materials Theory

Conventional mechanism for ferroelectricity:

paraelectric

ferroelectric

_

_

_

_

+

+

Curie Temperature

Materials Theory

Conventional mechanism for ferroelectricity:

paraelectric

ferroelectric

_

_

_

_

+

+

Curie Temperature

02/04/14

2

Materials Theory

Conventional mechanism for ferroelectricity:

paraelectric

ferroelectric

_

_

_

_

+

+

Ligand field stabilization of empty cation d orbitals by oxygen p electrons:

Materials Theory

Perturbation theory for structural distortions!Expand Hamiltonian as function of atomic distortion (normal coordinate), Q:

Then where!

1st-order JT Non-zero for orbitally

degenerate states centrosymmetric

always negative (relaxation of electron distribution);

want this to be large need a non-zero matrix

element for En close to E(0)

always positive (moving nuclei with

fixed electrons); want this to be small

Second-order Jahn-Teller effect

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Materials Theory

BaTiO3 (d0 ) Repulsive term small Energy-lowering term non-zero

CaMnO3 (d3 ) Repulsive term large Energy-lowering term 0 by symmetry

Perturbation theory

Materials Theory Materials Theory

D-MATL / Materials Theory

Multiferroics

Monday, January 13, 2014 6

ferroelectrics

ferroelastics ferromagnets

ε H

M

E

P

σ

+ - + -

+ - + -

) ) ) ) ) ) ) ) ) ) ) )

ε M

N S

P

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Materials Theory

Chemical contra-indication

Conventional ferroelectricity requires “d 0-ness” Conventional ferromagnetism requires some sort of localized electrons such as transition metal d electrons

!

Materials Theory

How to combine M and P?

either 1)  use an alternative mechanism for P

or 2)  use an alternative mechanism for M!

02/04/14

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Materials Theory

How to combine M and P?

either 1)  use an alternative mechanism for P

or 2)  use an alternative mechanism for M!

Materials Theory

Classification of known multiferroics

R.E. Newnham et al., Magnetoferroelectricity in Cr2BeO4, J. Appl. Phys. 49, 6088 (1978)

Magnetically driven e.g. Cr2BeO4

N. Ikeda et al., Ferroelectricity from iron valence ordering in the charge-frustrated system LuFe2O4, Nature 436, 1136 (2005)

Charge ordered e.g. LuFe2O4

Lone pair active e.g. BiMnO3, BiFeO3

Geometric ferroelectricity e.g. BaNiF4 C. Ederer and N.A. Spaldin, Electric-field switchable magnets: The case of BaNiF4, PRB 74, 020401(R) (2006)

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Materials Theory

Not very clever ways combine magnetism and ferroelectricity

magnetic and make this one ferroelectric

OR

ferroelectric let this ion be magnetic

Choose compounds (oxides) with 2 cations!

Materials Theory

Lone-pair active ferroelectrics

let this ion be magnetic

and make this one ferroelectric

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Materials Theory

Find: Calculate and measure P = 90 µC/cm2 (huge!) BUT: anti-ferromagnetic alignment of Fe3+ ions

Lone-pair active multiferroics: BiFeO3

Epitaxial BiFeO3 multiferroic thin film heterostructures, Wang, Spaldin, Ramesh et al., Science 299, 1719 (2003)

Idea: Ferroelectricity from the “stereochemically active lone pair” on Bi3+ (cf ammonia, NH3) Magnetism from a 3d transition metal (Fe3+)

Materials Theory

Lone-pair active multiferroics: BiMnO3

Ferromagnetic!

-5.E-04-4.E-04-3.E-04-2.E-04-1.E-040.E+001.E-042.E-043.E-044.E-045.E-04

-10000 -5000 0 5000 10000

Magnetic Field (Gauss)

Mag

netiz

atio

n (E

MU

)

A.M. Santos, S. Parashar, A.R. Raju, Y.S. Zhao, A.K. Cheetham and C.N.R. Rao, Evidence for the likely occurrence of magnetoferroelectricity in BiMnO3, Sol. Stat. Comm. 122, 49 (2002).

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Materials Theory

Lone-pair active multiferroics: BiMnO3 A.M. Santos et al., Orbital ordering as the determinant for ferromagnetism in biferroic BiMnO3, PRB 66, 064425 (2002).

Why is BiMnO3 ferromagnetic? Different strain gives different orbital ordering!

Materials Theory

BUT: anti-ferroelectric

Lone-pair active multiferroics: BiMnO3

P. Baettig, R. Seshadri and N. A. Spaldin, Antipolarity in ideal BiMnO3, JACS 129, 9854 (2007).

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Materials Theory

Magnetically-driven Ferroelectrics: Spiral magnets

T. Kimura et al., Magnetic control of ferroelectric polarization, Nature 426, 55 (2004)

e.g. chromium chrysoberyl magnetic structure at 4.5K is a cycloidal spiral without an inversion center spontaneous polarization around 1000 times smaller than in conventional ferroelectrics

Recently rediscovered in TbMnO3

R.E. Newnham et al., Magnetoferroelectricity in Cr2BeO4, J. Appl. Phys. 49, 6088 (1979)

“Magnetoferroelectrics might be termed the ultimate impropriety”

Materials Theory

Charge-ordered multiferroics N. Ikeda et al., Ferroelectricity from iron valence ordering in the charge-frustrated system LuFe2O4, Nature 436, 1136 (2005) e.g. LuFe2O4

Intriguing open question: Is magnetite below the Verwey transition a charge-ordered ferroelectric?

J. van den Brink and D. Khomskii, Multiferroicity due to charge ordering, arXiv:0803.2964

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Materials Theory

Geometric ferroelectrics

Tiltings in 3-dimensionally connected structures are non-polar

BUT 2-D connectivities can have polar tilt modes!!!

Materials Theory

Geometric ferroelectrics: BaNiF4

reference structure polar ground state

C. Ederer and N.A. Spaldin, Origin of ferroelectricity in the multiferroic barium fluorides BaMF4: A first principles study, Phys Rev B 74, 024102 (2006) C. Ederer and N.A. Spaldin, Electric-field switchable magnets: The case of BaNiF4, PRB 74, 020401(R) (2006)

02/04/14

11

Materials Theory

Is there a way to make the conventional ferroelectric mode and magnetism compatible?!

Is it possible to achieve this in systems with partially filled d-shells?!

Make this term small! Make this term large!

Reminder: Second-Order Jahn-Teller Effect

Materials Theory

Tailoring the 2nd order Jahn-Teller effect"We need non-zero matrix elements between the ground state and the low-lying excited

states (and their energy difference to be small)!

energy

DOS

O 2p

TM 3d

BaTiO3

ferroelectric

d0 band insulator

EF

Non-zero matrix elements requires that the symmetry of the two states be of opposite parity!

Products between p- and d- states do not vanish!

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Materials Theory

Tailoring the 2nd order Jahn-Teller effect"We need non-zero matrix elements between the ground state and the low-lying excited

states (and their energy difference to be small)!

energy

DOS

O 2p

TM 3d

DOS BaTiO3

CaMnO3

ferroelectric non-polar

d0 band insulator mott-insulator

energy

EF

Materials Theory

Other competing interactions"

perovskite tolerance factor! Cubic perovskites (t=1) are only found when stabilized by oversized A-site cations and/or TM—O π-bonding.!

Often t<1 and antiferrodistortive rotations (AFDs) of the oxygen octahedral occur and disfavor ferroelectricity!

these rotational modes can be tuned in thin films by changing the epitaxial strain state

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Materials Theory

A familiar d3 system, CaMnO3 "

[25]

Z*! Ca! Mn! O⊥! O‖!

Formal! + 2.00! + 4.00! – 2.00! – 2.00!DFT (GGA)!

+ 2.61! + 8.68! – 1.80! – 7.18!

CaMnO3 is not ferroelectric, but is predicted to be so when strained, or under ”negative pressure”

CaMnO3 has anomalous BECs suggesting that there is an underlying ferroelectric instability due to destabilizing dipolar interactions!

Ph. Ghosez, et al Phys. Rev. Lett. 102 117602 (2009)!

Materials Theory

How can we make negative pressure in the lab.? "Make perovskite BaMnO3 !

[26]

BaMnO3 is experimentally found in a hexagonal (P63/mmc) phase with face- and edge-shared oxygen polyhedra

1.34

1.44

1.61 increasing radius (Å)

Caveat

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14

Materials Theory

Structural instabilities"

Phonon dispersions calculated for the cubic perovskite structure!

Imaginary frequencies indicate structure is unstable with respect to atomic displacements

antiferrodistortive mode is stable

strong FE instability

octahedral distortions

Materials Theory

Calculated ground state"

orthorhombic (Amm2) We calculate a 12.8 µC/cm2 polarization along [101]

same polar phase as finite temperature BaTiO3

1.88 Å 1.95 Å

1.92 Å

c

b

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Materials Theory

Electronic Structure"Maximally-localized oxygen centered pz Wannier functions from the localization

of the full valence band in the centrosymmetric and polar structure!

In the polar structure, the pz orbital hybridizes with the d(x2–y2) orbital of the Mn atom!

non-polar polar

Materials Theory

Electronic Structure"

Born effective charges!Ba! Formal! + 2.00!

DFT (LSDA)!

+ 2.72!

Mn! Formal! + 4.00!DFT (LSDA)!

+ 9.17!

O! Formal! – 2.00!DFT (LSDA)!

– 3.97!

Mn-driven ferroelectricity

J.M. Rondinelli, A. Eidelson and N.A. Spaldin, Non-d0 Mn-driven ferroelectricity in antiferromagnetic BaMnO3, PRB 79, 205119 (2009)

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Materials Theory

Structures close to their limits of stability show divergent properties!"

CaMnO3 SrMnO3 BaMnO3

volume

Perovskite Hexagonal hexagonal

perovskite structure becomes polar

Since demonstrated experimentally in strained SrMnO3 (Fiebig et al.) and perovskite-structure(Sr,Ba)MnO3 (Tokura et al.)