# 16. Dielectrics and Ferroelectrics by j1jnZ0D

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```									16. Dielectrics and Ferroelectrics
Maxwell Equations
Polarization
Macroscopic Electric Field
Depolarization Field, E1
Local Electric Field at An atom
Lorentz Field, E2
Field of Dipoles Inside Cavity, E3
Dielectric Constant And Polarizability
Electronic Polarizability
Classical Theory
Examples
Structural Phase Transitions
Ferroelectric Crystals
Classification of Ferroelectric Crystals
Displacive Transitions
Soft Optical Phonons
Landau theory of the Phase Transition
Second-Order Transition
First-Order Transition
Antiferroelectricity
Ferroelectric Domains
Piezoelectricity
Maxwell Equations

 B  0                E  4 

B                   4    D
E                  H       J
c t                  c    c t
Polarization
Polarization P  dipole moment per unit volume.

Total dipole moment     p   qn rn
n

For a neutral system, p is independent of the choice of the coordinate origin.

H2 O

3p  r  r  r 2 p
Dipole field:        E r  
r5
Macroscopic Electric Field
E0 = external (applied) field                              e(r) = microscopic field
1
Macroscopic field     E  r0           dV e  r        VC = volume of crystal cell
VC

E due to a volume of uniform P is equal to that due to a surface charge density             nP
ˆ

For any point between the plates & far from the edges E1  4                 4 P

E  E0  E1  E0  4 P z
ˆ                     E1 = field due to ζ = n  P.
Depolarization Field, E1
If P is uniform, then   E  E0  E1              E1 = depolarization field

Inside an ellipsoidal body, P uniform → E1 uniform

Along the principal axes E1 j   N j Pj

Nj = depolarization factor

Nj
j     4

ellipsoid of revolution
PE              χ = dielectric susceptibility

Along a principal axis of an ellipsoid:   E  E0  E1  E0  N P


→        P    E0  N P              E0
1 N 
Local Electric Field at an Atom
In general, E  Eloc.

Consider a simple cubic crystal of spherical shape.
4
The macroscopic field in a sphere is                    E  E0  E1  E0              P
3
If all dipoles are equal to p  p z
ˆ                the dipole field at the center of the sphere is

3z 2  rj2                    2 z 2  x2  y 2
Edipole  p                        p
j                              j    j     j

j       rj5                  j                rj5
x2           y2                z2
Cubic symmetry →            r
j
j
5

j   r
j
5

j   r
j
5
→   E dipole  0   →     Eloc  E0  E
j               j                j

For an abitrary symmetry                  Eloc  E0  E1  E2  E3

E2 = Lorentz cavity field
(due to charges on surface of cavity)
E3 = field of atoms inside cavity

E1 + E2 = field of body with hole.
E1 + E2 + E3= field of all other atoms at one atom.

3  p j  r j  r j  rj2 p j
E1  E2  E3  
j 0                rj5

Sites 10a (~50A) away can be replaced by 2 surface integrals:
1 over the outer ellipsoidal surface,
the other over the cavity defining E2 .
Lorentz Field, E2


P cos 
E2    2 asin    a d        2
cos 
0
a
4
      P       E1
3

         E1  E2  0
Field of Dipoles inside Cavity, E3

E3 is only field that depends on crystal structure.

For cubic crystals     E3  0

4        4
→        Eloc  E0  E1       P  E    P           Lorentz relation
3         3
Dielectric Constant and Polarizability
E  4 P
For an isotropic / cubic medium, ε is a scalar:                                  1  4 
E
P   DE     1
              
E   4 E    4

For a non-cubic medium, ε & χ are tensors:

P    E                      4  

Polarizability α of an atom:          p   Eloc                  α is in general a tensor

Polarization: P   N j p j   N j  j Eloc  j 
j            j

    4 
For cubic medium, Lorentz relation applies :                   P   
j
N j  j E 

P
3 
N       j   j
                                    1 4
N
j
4                     →                              j          Clausius-Mossotti relation
N                      2 3                 j
1               j   j                         j
3   j
Electronic Polarizability

Dipolar: re-orientation of
molecules with permanent dipoles

Ionic: ion-ion displacement

Electronic: e-nucleus displacement

In heterogeneous materials, there is also an interfacial polarization.
At high frequencies, electronic contribution dominates.
  1 4                                   n2  1
e.g., optical range:         
 2 3
N
j
j    j  electronic     2
n 2
Classical Theory of Electronic Polarizability
Bounded e subject to static Eloc :         m  x  0 x   eEloc
2

e Eloc
m 0 2

p     e x    e2
Static electronic polarizability:             el              
Eloc   Eloc   m 02

Bounded e subject to oscillatory Eloc :            m  x  0 x   eEloc sin  t
2

e Eloc
Oscillatory solution :   x  x0 sin  t                 x0 
m  2  0 
2

e2
Electronic polarizability:         el  
m 0   2 
2

e2              fi j
Quantum theory:        el  
m

j
2
2                 f i j = oscillator strength of dipole
ij
transition between states i & j.
Structural Phase Transitions
At T = 0, stable structure A has lowest free energy F = U for a given P.
High P favors close-packing structures which tend to be metallic.
E.g., H & Xe becom metallic under high P.

Let B has a softer (lower ω) phonon spectrum than A.
→ SB > SA due to greater phonon occupancy for B.
→  TC s.t. FB = UB –T SB > FA = UA –T SA  T > TC
( phase transition A → B unless TC > Tmelt )
FB (TC ) = FA (TC )
Near TC , transition can be highly stress sensitive.

Ferroelectrics: spontaneous P.
• Unusual ε(T).
• Piezoelectric effect.
• Pyroelectric effect.
• Electro-optical effects such as optical frequency doubling.
Ferroelectric Crystals

Ferroelectric  TC → Paraelectric

Ferroelectric : P vs E plot shows hysteresis.

Pyroelectric effects (P  T ) are often
found in ferroelectrics where P is not
affected by E less than the breakdown field.

E.g., LiNbO3 is pyroelectric at 300K.
High TC = 1480K.
Large saturation P = 50 μC/cm2 .
Can be “poled”
(given remanent P by E at T >1400K).

PbTiO3
Classification of Ferroelectric Crystals
2 main classes of ferroelectrics:
• order-disorder: soft (lowest ωTO ) modes diffusive at transition.
e.g., system with H-bonds: KH2PO4 .                              Most are in between
• displacive: soft modes can propagate at transition.
e.g., ionic crytsls with perovskite, or ilmenite structure.

Order-disorder

TC nearly doubled on H→D.
Due to quantum effect involving mass-dependent de Broglie wavelength.
n-diffraction → for T < TC , H+ distribution along H-bond asymmetric.

Displacive

T > TC                                                                T < TC : displaced
BaTiO3
At 300K, PS = 8104 esu cm–2 .
→       p  510–18 esu cm
VC = (4 10–8 )3 = 64 10–24 cm3.

Moving Ba2+ & Ti4+ w.r.t. O2– by δ = 0.1A gives p /cell = 6e δ  310–18 esu cm

In LiNbO3, δ is 0.9A for Li & 0.5A for Nb → larger p.
Displacive Transitions
2 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 .

8
1
3
N    j   j
3
                                                      Nj j 
j
C-M relation:
4                       Catastophe condition:
4
1
3
N
j
j   j                                j
8
4
1
3
N   j   j
3  6s   1
 N j  j  1  3s                   j
→
4                                   for s → 0
3    j                          1
3
N
j
j   j         3s     s

T  TC              
s            →                   (paraelectric)
               T  TC
Soft Optical Phonons
TO    
2
ωTO → 0  ε(0) → 
LST relation        
LO   0 
2
no restoring force: crystal unstable

E.g., ferroelectric BaTiO3 at 24C has ωTO = 12 cm–1 .
1                        TO  T  TC
 T  TC     →                          if ωLO is indep of T
2
Near TC ,
  0

SrTiO3                                                                            SbSI
from n scatt                                                                      from Raman scatt
Landau Theory of the Phase Transition
Landau free energy density:

1
F  P; T , E    E  P  g 0         g 2 j P 2 j  E  P  g  1 g P 2  1 g P 4  1 g P 6 
j 1   2j                         0
2
2
4
4
6
6

• 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

0  F  P; T , E  E  g2P  g4P3  g6P5 

Transition to ferroelectric is facilitated by setting      g2   T  T0           0 , T0  TC

(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.
Second-Order Transition
0  E   T  T0  P  g4P3  g6P5 

For g4 > 0, terms g6 or higher bring no new features & can be neglected.

E=0 →         0   T  T0  P  g4P3      → PS = 0 or      PS2         T0  T 
g4
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 , PS           T0  T     minimizes F ( T, 0 ) (ferroelectric phase).
g4

LiTaO3
First-Order Transition
0  E   T  T0  P  g4P3  g6P5 

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

E=0 →             0   T  T0  P  g4 P3  g6P5

1 
→ PS = 0 or           PS2           g4      g 4  4 g 6 T  T0  
2

2 g6                                  

BaTiO3 (calculated)

For E  0 & T > TC , g4 & higher terms can be neglected:             E   T  T0  P
4 P         4
  1         1                        T0 = TC for 2nd order trans.
E         T  T0             T0 < TC for 1st order trans.
Fundamental types of structural phase
transitions from a centrosymmetric prototpe
Perovskite
Widely used as ceramic piezoelectrics.
Ferroelectric Domains

Atomic displacements                                          Domains with
of oppositely polarized                                       180 walls
domains.

BaTiO face  c axis.
Ea // c axis.
Piezoelectricity
Ferroelectricity → Piezoelectricity (not vice versa)

BaTiO3 :
d = 10−5 cm/statvolt

Unstressed

Pd  E                    ζ   =   stress (tensor)                                             e 
d   =   piezoelectric constant (tensor)                      di    
e  sd E                   χ   =   dielectric susceptibility                                    Ei 
e   =   elastic compliance constant (tensor)                    α = 1,∙∙∙, 6

A+3 B3−
PiezoE not FerroE
e.g., SiO2
d  10−7 cm/statvolt

Unstressed: 3-fold symmetry
PVF2 films are flexible & often used as ultrsonic transducers

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