16. Dielectrics and Ferroelectrics

Maxwell EquationsPolarization

Macroscopic Electric FieldDepolarization Field, E1

Local Electric Field at An atomLorentz Field, E2

Field of Dipoles Inside Cavity, E3

Dielectric Constant And PolarizabilityElectronic PolarizabilityClassical TheoryExamples

Structural Phase TransitionsFerroelectric Crystals

Classification of Ferroelectric CrystalsDisplacive Transitions

Soft Optical PhononsLandau theory of the Phase TransitionSecond-Order TransitionFirst-Order TransitionAntiferroelectricityFerroelectric DomainsPiezoelectricity

Maxwell Equations

4 E0 B

c t

B

E 4

c c t

D

H J

Polarization

Polarization P dipole moment per unit volume.

n nn

qp rTotal dipole moment

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

Dipole field: 2

5

3 r

r

p r r pE r

H2 O

Macroscopic Electric Field

E0 = external (applied) field

Macroscopic field 0

1

C

dVV

E r e r VC = volume of crystal cell

e(r) = microscopic field

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

For any point between the plates & far from the edges 1 4E 4 P

0 1 E E E 0 ˆ4 P E z E1 = field due to σ = n P.

Depolarization Field, E1

Inside an ellipsoidal body, P uniform → E1 uniform

0 1 E E E E1 = depolarization fieldIf P is uniform, then

Along the principal axes 1 j j jE N P

Nj = depolarization factor

4jj

N

ellipsoid of revolution

P E χ = dielectric susceptibility

Along a principal axis of an ellipsoid: 0 1E E E 0E N P

→ 0P E N P 01

EN

Local Electric Field at an Atom

Consider a simple cubic crystal of spherical shape.

0 1 0

4

3

E E E E P

If all dipoles are equal to

The macroscopic field in a sphere is

In general, E Eloc.

ˆpp z the dipole field at the center of the sphere is

2 2

5

3 j jdipole

j j

z rp

r

E

2 2 2

5

2 j j j

j j

z x yp

r

Cubic symmetry →2 2 2

5 5 5

j j j

j j jj j j

x y z

r r r → 0dipole E → 0loc E E E

For an abitrary symmetry 0 1 2 3loc E E E E E

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.

2

1 2 3 50

3 j j j j j

j j

r

r

p r r pE E E

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

2 20

cos2 sin cosa a d

a

P

E

4

3

P 1 E

1 2 0 E E

Field of Dipoles inside Cavity, E3

E3 is only field that depends on crystal structure.

For cubic crystals 3 0E

→ 0 1

4

3loc

E E E P

4

3

E P Lorentz relation

Dielectric Constant and Polarizability

For an isotropic / cubic medium, ε is a scalar: 4E P

E

1 4

P

E

4

D E

E

1

4

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

P E 4

Polarizability α of an atom: locp E α is in general a tensor

jjj

NP p j ojj

l c jN EPolarization:

For cubic medium, Lorentz relation applies :4

3jjj

N

P E P

41

3

jj

j

j

jj

N

N

→ 1 4

2 3 jj

jN

Clausius-Mossotti relation

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.

e.g., optical range: 1 4

2 3 jj

j elec o cN tr ni

2

2

1

2

n

n

Classical Theory of Electronic Polarizability

20 locm x x eE Bounded e subject to static Eloc :

Steady state:20

loce Ex

m

Static electronic polarizability: loc

pel

E

2

20

e

m

loc

e x

E

20 sinlocm x x eE t Bounded e subject to oscillatory Eloc :

0 2 20

loce Ex

m

Oscillatory solution :

0 sinx x t

Electronic polarizability: 2

2 20

eel

m

Quantum theory: 2

2 2

i j

i jj

eel

m

f

f i j = oscillator strength of dipole 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

PbTiO3

Ferroelectric : P vs E plot shows hysteresis.

Ferroelectric TC → Paraelectric

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).

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 .• displacive: soft modes can propagate at transition.

e.g., ionic crytsls with perovskite, or ilmenite structure.

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.

T > TC T < TC : displaced

Most are in between

Order-disorder

Displacive

BaTiO3

At 300K, PS = 8104 esu cm–2 .VC = (4 10–8 )3 = 64 10–24 cm3.

→ p 510–18 esu cm

Moving Ba2+ & Ti4+ w.r.t. O2– by δ = 0.1A gives p /cell = 6e δ 310–18 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 .

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 Phonons

LST 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:

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).

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.

Fundamental types of structural phase transitions from a centrosymmetric prototpe

Perovskite Lead zirconate-lead titanate (PZT) systemWidely used as ceramic piezoelectrics.

Ferroelectric Domains

Atomic displacements of oppositely polarized domains.

Domains with 180 walls

BaTiO face c axis.Ea // c axis.

Piezoelectricity

Ferroelectricity → Piezoelectricity (not vice versa)

Unstressed

d P E σ = stress (tensor)d = piezoelectric constant (tensor)χ = dielectric susceptibilitye = elastic compliance constant (tensor)

d e s Ei

i

ed

E

α = 1,∙∙∙, 6

Unstressed: 3-fold symmetry

A+3 B3− PiezoE not FerroE

e.g., SiO2

d 10−7 cm/statvolt

BaTiO3 :d = 10−5 cm/statvolt

PVF2 films are flexible & often used as ultrsonic transducers