16.3 Ferroelectricity
455
16.3.3 Phase Transition
In the case of ferroelectrics, the order parameter for the Ginzburg–Landau theory of phase transitions
is the spontaneous polarization P. The free energy F of the ferroelectric crystal is written in terms of
the free energy of the paraelectric phase F 0 and is expanded in powers of P (here up to P
6 ) as
F = F 0 +
1
2
α P
2
+
1
4
β P
4
+
1
6
γ P
6
.
(16.2)
In this equation, we have neglected effects due to charge carriers, an external electric field or external
stresses and we assume homogeneous polarization. In order to obtain a phase transition, it has to be
assumed that α has a zero at a certain temperature T C and we assume (expanding only to the linear
term)
α = α 0 (T − T C ) .
(16.3)
16.3.3.1 Second-Order Phase Transition
For modeling a second-order phase transition, we set γ = 0. Thus, the free energy has the form
(Fig. 16.7a)
F = F 0 +
1
2
α P
2
+
1
4
β P
4
.
(16.4)
The equilibrium condition with regard to the free energy yields a minimum for
∂ F
∂ P
= α P + β P
3
= 0
(16.5a)
∂
2 F
∂ P 2 = α + 3 β P
2
> 0 .
(16.5b)
Equation (16.5a) yields two solutions. P = 0 corresponds to the paraelectric phase. P
2
= −α/β is the
spontaneous polarization in the ferroelectric phase. The condition from (16.5b) yields that α > 0 in the
paraelectric phase, while α < 0 in the ferroelectric phase. Also, β > 0 below the Curie temperature
(β is assumed to be temperature independent in the following). Using (16.3), the polarization is given
as (Fig. 16.7b)
P
2
=
α 0
β
(T − T C ) .
(16.6)
Therefore, the entropy S = −
∂ F
∂ T
and the discontinuity C p of the heat capacity C p = T
∂ S
∂ T
p
at the
Curie point T C are given by
S = S 0 +
α
2
0
β
(T − T C )
(16.7a)
C p =
α
2
0
β
T C ,
(16.7b)
with S 0 = −
∂ F 0
∂ T
being the entropy of the paraelectric phase. This behavior is in accordance with a
second-order phase transition with vanishing latent heat (continuous entropy) and a discontinuity of
the heat capacity. The dielectric function in the paraelectric phase is ∝ 1/α and in the ferroelectric
phase ∝ −1/α. The latter relation is usually written as the Curie–Weiss law
455
16.3.3 Phase Transition
In the case of ferroelectrics, the order parameter for the Ginzburg–Landau theory of phase transitions
is the spontaneous polarization P. The free energy F of the ferroelectric crystal is written in terms of
the free energy of the paraelectric phase F 0 and is expanded in powers of P (here up to P
6 ) as
F = F 0 +
1
2
α P
2
+
1
4
β P
4
+
1
6
γ P
6
.
(16.2)
In this equation, we have neglected effects due to charge carriers, an external electric field or external
stresses and we assume homogeneous polarization. In order to obtain a phase transition, it has to be
assumed that α has a zero at a certain temperature T C and we assume (expanding only to the linear
term)
α = α 0 (T − T C ) .
(16.3)
16.3.3.1 Second-Order Phase Transition
For modeling a second-order phase transition, we set γ = 0. Thus, the free energy has the form
(Fig. 16.7a)
F = F 0 +
1
2
α P
2
+
1
4
β P
4
.
(16.4)
The equilibrium condition with regard to the free energy yields a minimum for
∂ F
∂ P
= α P + β P
3
= 0
(16.5a)
∂
2 F
∂ P 2 = α + 3 β P
2
> 0 .
(16.5b)
Equation (16.5a) yields two solutions. P = 0 corresponds to the paraelectric phase. P
2
= −α/β is the
spontaneous polarization in the ferroelectric phase. The condition from (16.5b) yields that α > 0 in the
paraelectric phase, while α < 0 in the ferroelectric phase. Also, β > 0 below the Curie temperature
(β is assumed to be temperature independent in the following). Using (16.3), the polarization is given
as (Fig. 16.7b)
P
2
=
α 0
β
(T − T C ) .
(16.6)
Therefore, the entropy S = −
∂ F
∂ T
and the discontinuity C p of the heat capacity C p = T
∂ S
∂ T
p
at the
Curie point T C are given by
S = S 0 +
α
2
0
β
(T − T C )
(16.7a)
C p =
α
2
0
β
T C ,
(16.7b)
with S 0 = −
∂ F 0
∂ T
being the entropy of the paraelectric phase. This behavior is in accordance with a
second-order phase transition with vanishing latent heat (continuous entropy) and a discontinuity of
the heat capacity. The dielectric function in the paraelectric phase is ∝ 1/α and in the ferroelectric
phase ∝ −1/α. The latter relation is usually written as the Curie–Weiss law