Displacive TransitionsSoft Optical PhononsLandau Theory of the Phase TransitionSecond-Order TransitionFirst-Order Transition

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Classic example – BaTiO3 which exhibits ferroelectricity

Figure adapted from Callister, Materials science and engineering, 7 th Ed.http://www.camsoft.co.kr

B (Ti) sits inside an octahedral cage of Oxygens

BaTiO3

Perovskites – ABO3

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SrTiO3

TiO

Sr

Sr2+ O2-Ti4+

http://www.camsoft.co.kr

ABO3

a

A

B

O

B sites are octahedrally bonded by oxygens

For an undistorted cube:

2

2

2

A O

B O

O O

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SrTiO3

Ideal Perovskite Structures

Displacive Phase Transitions

A

B

OIonic radii never match ideal cubic requirements.

A site atoms smaller than hole:

In displacive phase transitions the atoms only change position slightly.

Distortion of octahedra

LaMnO3

Most perovskite structures are distorted due to the ionic radii of

the cations and distortions caused by the local crystal fields

and electron interactions

- Temperature Dependent

European Synchrotron Radiation Facility, Research Highlights, 2001

Structural changes can induce other phenomena

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SrTiO3 - Tc=105K

Antiferrodistortive transition – unit cell doubled

Displacive TransitionsBaTiO3

Centrosymmetric

Non-centrosymmetric

Displacive Transitions2 viewpoints on displacive transitions:• Polarization catastrophe

( Eloc caused by u is larger than elastic restoring force ).• Condensation of TO phonon

(t-indep displacement of finite amplitude)Happens when ωTO = 0 for some q 0. ωLO > ωTO & need not be considered .

In perovskite structures, environment of O2– ions is not cubic → large Eloc.→ displacive transition to ferro- or antiferro-electrics favorable.

Catastophe theory:

Let Eloc = E + 4 π P / 3 at all atoms.In a 2nd order phase transition, there is no latent heat.The order parameter (P) is continuous at TC .

81

34

13

j jj

j jj

N

N

C-M relation:

Catastophe condition:3

4j jj

N

81

34

13

j jj

j jj

N

N

4

33

1j jj

N s → 3 6

3

s

s

1

s for s → 0

CsT T

→CT T

(paraelectric)

Soft Optical PhononsLST relation

2

2 0TO

LO

ωTO → 0 ε(0) →

no restoring force: crystal unstable

E.g., ferroelectric BaTiO3 at 24C has ωTO = 12 cm–1 .

Near TC , 1

0 CT T

→ 2TO CT T if ωLO is indep of T

SrTiO3

from n scatt

SbSIfrom Raman scatt

Landau Theory of the Phase Transition

Landau free energy density at 1D:

2 4 60 2 4 6

1 1 1

2 4 6g g g g E P P P P

Comments:• Assumption that odd power terms vanish is valid if crystal has center of inversion.• Power series expansion often fails near transition (non-analytic terms prevail) . e.g., Cp of KH2PO4 has a log singularity at TC .

The Helmholtz free energy F(T, E) is defined by

3 52 4 60 ; , g g gF T P PE PP E

Transition to ferroelectric is facilitated by setting 2 0g T T 00 , CT T

(This T dependence can be explained by thermal expansion & other anharmonic effects )

g2 ~ 0+ → lattice is soft & close to instability.g2 < 0 → unpolarized lattice is unstable.

20 2

1

; ,1

2j

jj

g gj

F T

PE PP E

Second-Order Transition

For g4 > 0, terms g6 or higher bring no new features & can be neglected.

3 50 4 60 T T g g P PE P

E = 0 → 30 40 T T g P P → PS = 0 or

2

04

S T Tg

P

Since γ , g4 > 0, the only real solution when T > T0 , is PS = 0 (paraelectric phase).This also identifies T 0 with TC .

For T < T0 ,

04

SP T Tg

minimizes F ( T, 0 ) (ferroelectric phase).

Spontaneous polarization versus temperature in second order transition

Temperature variation of the polar axis static electric constant of LiTaO3

First-Order Transition

For g4 < 0, the transition is 1st order and term g6 must be retained.

3 50 4 60 T T g g P PE P

E = 0 → 3 50 4 60 T T g g P P P

→ PS = 0 or

2 24 4 6 0

6

14

2S g g g T Tg

P

BaTiO3 (calculated)

For E 0 & T > TC , g4 & higher terms can be neglected: 0T T E P

0

4 41 1

P

E T T

T0 = TC for 2nd order trans.T0 < TC for 1st order trans.

Landau free energy function versus(polarization)2 in a first order transition

n = 1 First Order

0

0

(G)0

(T)CT T

1

1

(G)

(T) CP

HS

T

G H T S

dG VdP SdT

0G

(G)

(T)P

S

Finite discontinuity

2

2P

PP

Cd G dS

dT dT T

1

1

(G)0

(T) CP

HS

T

Sche

mat

ics

2

2P

P CP

d G dS C

dT dT T

0G

1

1

(G)0

(T) CP

HS

T

n = 2 Second Order

1

1

(G)0

(T)C

CT T

HS

T

2

2

(G) 10

(T)P

PC CP

H C

T T T

G H T S dG VdP SdT

(G)

(T)P

S

Finite discontinuity

Second derivative is CP

Phase Transformations: Examples from Ti and Zr Alloys, S. Banerjee and P. Mukhopadhyay, Elsevier, Oxford, 2007

Schematics