16.3 Ferroelectricity
457
(a)
(b)
25
20
15
10
5
0
P (10 C/m )
S
2
-2
20
40 50 60 70 80 90 100 110
30
BaTiO 3
Fig. 16.8 a Schematic plot of the free energy versus spontaneous polarization for a first-order phase transition. The
lowest curve is for α = 0, the others are for α > 0. b Spontaneous polarization in BaTiO 3 as a function of temperature
exhibiting a first-order phase transition. The dashed line is a guide to the eye. Adapted from [1498]
For T ≤ T 1 the absolute minimum of the free energy is reached for finite polarization P > P 0 .
However, between F(P = 0) and the minimum of the free energy an energy barrier (second lowest
curve in Fig. 16.8) is present for T close to T 1 . The energy barrier disappears at the Curie–Weiss
temperature T 0 . At the phase transition temperature, the entropy has a discontinuity
= α 0 P
2
0 ,
(16.13)
that corresponds to a latent heat = T Another property of the first-order phase transition is
the occurrence of hysteresis in the temperature interval between T 1 and T 0
≈ T 1 − T 0 =
1
4α 0
β
2
γ
,
(16.14)
in which an energy barrier is present to hinder the phase transition. For decreasing temperature, the
system tends to remain in the paraelectric phase. For increasing temperature, the system tends to remain
in the ferroelectric phase. Such behavior is observed for BaTiO 3 , as shown in Fig. 16.4b.
16.3.4 Domains
Similar to ferromagnets, ferroelectrics form domains with different polarization directions in order
to minimize the total energy by minimizing the field energy outside the crystal. The polarization can
have different orientations, 6 directions for P along 100 (tetragonal phase), 12 directions for P along
110 (orthorhombic phase) and 8 directions for P along 111 (rhombohedral phase). In Fig. 16.9,
such domains are visualized for BaTiO 3 . Due to the restricted geometry, domain formation in thin
films is different from that in bulk material.
Domains can also be artificially created by so-called poling. The ferroelectric semiconductor is
heated to the paraelectric phase. With electrodes, appropriate electric fields are applied and the material
is cooled. The polarization is then frozen in the ferroelectric phase. The domains of a periodically
poled structure in LiNbO 3 (PPLN) are shown in Fig. 16.10b. The nonlinear optical properties in such
structures can be used for efficient second harmonic generation (SHG).
457
(a)
(b)
25
20
15
10
5
0
P (10 C/m )
S
2
-2
20
40 50 60 70 80 90 100 110
30
BaTiO 3
Fig. 16.8 a Schematic plot of the free energy versus spontaneous polarization for a first-order phase transition. The
lowest curve is for α = 0, the others are for α > 0. b Spontaneous polarization in BaTiO 3 as a function of temperature
exhibiting a first-order phase transition. The dashed line is a guide to the eye. Adapted from [1498]
For T ≤ T 1 the absolute minimum of the free energy is reached for finite polarization P > P 0 .
However, between F(P = 0) and the minimum of the free energy an energy barrier (second lowest
curve in Fig. 16.8) is present for T close to T 1 . The energy barrier disappears at the Curie–Weiss
temperature T 0 . At the phase transition temperature, the entropy has a discontinuity
= α 0 P
2
0 ,
(16.13)
that corresponds to a latent heat = T Another property of the first-order phase transition is
the occurrence of hysteresis in the temperature interval between T 1 and T 0
≈ T 1 − T 0 =
1
4α 0
β
2
γ
,
(16.14)
in which an energy barrier is present to hinder the phase transition. For decreasing temperature, the
system tends to remain in the paraelectric phase. For increasing temperature, the system tends to remain
in the ferroelectric phase. Such behavior is observed for BaTiO 3 , as shown in Fig. 16.4b.
16.3.4 Domains
Similar to ferromagnets, ferroelectrics form domains with different polarization directions in order
to minimize the total energy by minimizing the field energy outside the crystal. The polarization can
have different orientations, 6 directions for P along 100 (tetragonal phase), 12 directions for P along
110 (orthorhombic phase) and 8 directions for P along 111 (rhombohedral phase). In Fig. 16.9,
such domains are visualized for BaTiO 3 . Due to the restricted geometry, domain formation in thin
films is different from that in bulk material.
Domains can also be artificially created by so-called poling. The ferroelectric semiconductor is
heated to the paraelectric phase. With electrodes, appropriate electric fields are applied and the material
is cooled. The polarization is then frozen in the ferroelectric phase. The domains of a periodically
poled structure in LiNbO 3 (PPLN) are shown in Fig. 16.10b. The nonlinear optical properties in such
structures can be used for efficient second harmonic generation (SHG).