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Displacive Transitions
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Transcript of Displacive Transitions
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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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web.uniovi.es/qcg/vlc/luana.htm
SrTiO3
TiO
Sr
Sr2+ O2-Ti4+
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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
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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
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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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web.uniovi.es/qcg/vlc/luana.html
SrTiO3 - Tc=105K
Antiferrodistortive transition – unit cell doubled
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Displacive TransitionsBaTiO3
Centrosymmetric
Non-centrosymmetric
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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
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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)
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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
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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
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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
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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
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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
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Sche
mat
ics
2
2P
P CP
d G dS C
dT dT T
0G
1
1
(G)0
(T) CP
HS
T
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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
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Phase Transformations: Examples from Ti and Zr Alloys, S. Banerjee and P. Mukhopadhyay, Elsevier, Oxford, 2007
Schematics
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