16.3 Ferroelectricity
453
(a)
cubic
tetragonal
c
(a c)
a=b
T c
0.415
0.410
0.405
0.400
0.395
0.390
0.385
T (°C)
PbTiO 3
(b)
20
15
10
5
0
P (10 C/m )
S
2
-
2
-50
BaTiO 3
-100
-150
0
0
5
150
100
-200
Fig. 16.4 a Cell parameters of PbTiO 3 as a function of temperature. Adapted from [1493]. b Phase transitions of BaTiO 3
as a function of temperature. The spontaneous polarization P S points along 100 110 and 111 in the tetragonal
(C 4v ), orthorhombic (C 2v ) and trigonal (C 3v , rhombohedral) phase, respectively. Adapted from [1494]
16.3 Ferroelectricity
Ferroelectric semiconductors exhibit a spontaneous polarization in the ferroelectric phase and zero
spontaneous polarization in the paraelectric phase. As a function of temperature, the ferroelectric
material undergoes a phase transition from the high-temperature paraelectric phase into the ferroelectric
phase. There can be further phase transitions between different ferroelectric phases that differ in the
direction of the polarization. The literature until 1980 is summarized in [1488]. A more recent treatment
can be found in [1489, 1490].
PbTiO 3 has perovskite structure (cf. Sect. 3.4.10). It exhibits a phase transition at T C = 490
◦ C from
the cubic into the (ferroelectric) tetragonal phase as shown in Fig. 16.4a. Mostly the cell symmetry
changes, while the cell volume remains almost constant. A more complicated situation arises for
BaTiO 3 . At 120
◦ C the transition into the ferroelectric phase occurs (Fig. 16.4b) that is tetragonal with
the polarization in the [100] direction. At −5
◦ C and −90
◦ C transitions occur into an orthorhombic
and a rhombohedral (trigonal) phase, respectively. The largest polarization is caused by a displacement
of the negatively (O) and positively (Ba, Ti) charged ions of the unit cell by δ ≈ 0.02 nm (Fig. 16.5).
Such an origin of the spontaneous polarization is called a displacement transition.
2
16.3.1 Materials
A large class of ferroelectric semiconductors are of the type ABO 3 , where A stands for a cation with
larger ionic radius and B for an anion with smaller ionic radius. Many ferroelectrics have perovskite
(CaTiO 3 ) structure. They are A
2+ B
4+ O
2−
3 , e.g. (Ba,Ca,Sr) (Zi,Zr)O 3 or A
1+ B
5+ O
2−
3 , e.g. (Li,Na,K)
(Nb,Ta)O 3 . Ferroelectrics can also be alloyed. Alloying in the B component yields, e.g. PbTi x Zr 1−x O 3
also called PZT. PZT is widely used for piezoelectric actuators. Also, alloying in the A component is
possible, e.g. Ba x Sr 1−x TiO 3 .
2 The widely accepted model for such ferroelectric is that the basic displacement occurs into the 111 at low
temperature. The three higher symmetry phases at higher temperature are the result of 2 (orthorhombic), 4 (tetragonal)
or 8 (cubic) allowed 111 orientations which make the macroscopically averaged polarization appear in 110- or
100 or vanishing altogether, respectively [1491, 1492].
453
(a)
cubic
tetragonal
c
(a c)
a=b
T c
0.415
0.410
0.405
0.400
0.395
0.390
0.385
T (°C)
PbTiO 3
(b)
20
15
10
5
0
P (10 C/m )
S
2
-
2
-50
BaTiO 3
-100
-150
0
0
5
150
100
-200
Fig. 16.4 a Cell parameters of PbTiO 3 as a function of temperature. Adapted from [1493]. b Phase transitions of BaTiO 3
as a function of temperature. The spontaneous polarization P S points along 100 110 and 111 in the tetragonal
(C 4v ), orthorhombic (C 2v ) and trigonal (C 3v , rhombohedral) phase, respectively. Adapted from [1494]
16.3 Ferroelectricity
Ferroelectric semiconductors exhibit a spontaneous polarization in the ferroelectric phase and zero
spontaneous polarization in the paraelectric phase. As a function of temperature, the ferroelectric
material undergoes a phase transition from the high-temperature paraelectric phase into the ferroelectric
phase. There can be further phase transitions between different ferroelectric phases that differ in the
direction of the polarization. The literature until 1980 is summarized in [1488]. A more recent treatment
can be found in [1489, 1490].
PbTiO 3 has perovskite structure (cf. Sect. 3.4.10). It exhibits a phase transition at T C = 490
◦ C from
the cubic into the (ferroelectric) tetragonal phase as shown in Fig. 16.4a. Mostly the cell symmetry
changes, while the cell volume remains almost constant. A more complicated situation arises for
BaTiO 3 . At 120
◦ C the transition into the ferroelectric phase occurs (Fig. 16.4b) that is tetragonal with
the polarization in the [100] direction. At −5
◦ C and −90
◦ C transitions occur into an orthorhombic
and a rhombohedral (trigonal) phase, respectively. The largest polarization is caused by a displacement
of the negatively (O) and positively (Ba, Ti) charged ions of the unit cell by δ ≈ 0.02 nm (Fig. 16.5).
Such an origin of the spontaneous polarization is called a displacement transition.
2
16.3.1 Materials
A large class of ferroelectric semiconductors are of the type ABO 3 , where A stands for a cation with
larger ionic radius and B for an anion with smaller ionic radius. Many ferroelectrics have perovskite
(CaTiO 3 ) structure. They are A
2+ B
4+ O
2−
3 , e.g. (Ba,Ca,Sr) (Zi,Zr)O 3 or A
1+ B
5+ O
2−
3 , e.g. (Li,Na,K)
(Nb,Ta)O 3 . Ferroelectrics can also be alloyed. Alloying in the B component yields, e.g. PbTi x Zr 1−x O 3
also called PZT. PZT is widely used for piezoelectric actuators. Also, alloying in the A component is
possible, e.g. Ba x Sr 1−x TiO 3 .
2 The widely accepted model for such ferroelectric is that the basic displacement occurs into the 111 at low
temperature. The three higher symmetry phases at higher temperature are the result of 2 (orthorhombic), 4 (tetragonal)
or 8 (cubic) allowed 111 orientations which make the macroscopically averaged polarization appear in 110- or
100 or vanishing altogether, respectively [1491, 1492].