52
3 Crystals
3.4.10 Perovskite Structure
The perovskite structure for ABO 3 materials (calcium titanate, CaTiO 3 , space group 62, Pnma)
(Fig. 3.27) is relevant for ferroelectric semiconductors (cf. Sect. 16.3). It is cubic with the Ca (or
Ba, Sr) ions (charge state 2+) on the corners of the cube, the O ions (2−) on the face centers and
the Ti (4+) in the body center. The lattice is simple cubic, the base is Ca at (0,0,0), O at (1/2,1/2,0),
(1/2,0,1/2) and (0,1/2,1/2) and Ti at (1/2,1/2,1/2). The ferroelectric polarization is typically evoked by
a shift of the negatively and positively charged ions relative to each other. LaAlO 3 (lanthanium aluminate) is available as substrate material (space group 226, Fm ¯
3c [227]). Perovskites are also important
for materials for high temperature superconductivity.
Another group of ABX 3 perovskites is formed by halogen atoms (Cl, Br, I) instead of oxygen
on the X-site (in −1 charge state). The B-site (former Ti-position) is taken up by lead (Pb) or other
elements of same +2 valence. The A-site (former Ba-position) is populated with an organic molecule
(+1) such as CH 3 NH 3 (methylammonium, or MA) [228] or HC(NH 2 ) 2 (formamidinium, or FA). In
a purely inorganic version of such halogen perovskite, the A-site site is populated with Cs [229]. In
order to avoid Pb, CsSnX 3 has been investigated. Various phases are observed next to the cubic phase
(α-CsPbI 3 ) such as orthorhombic (δ-CsPbBr 3 ) or tetragonal [230]. Also, angular distortions of the
PbX 6 octahedrons are common and depend on material composition (Fig. 3.28).
The phase stability depends on the octahedral factor μ = R B /R X and the tolerance factor t,
t =
R A + R X
√
2(R B + R X )
,
(3.12)
with R A , R B and R X denoting the radius of the A-site and B-site cation, and X-site anion, respectively. The stability of 138 different halide perovskites has been calculated in terms of μ and t [231]
(Fig. 3.29) and stability criteria have been given. A stable cubic phase requires 0.44 < μ < 0.9 and 0.8
≤ t ≤ 1 [232].
Table 3.5 Lattice parameters a, c, and u of some delafossite compounds. Theoretical values are shown with asterisk.
Data from [226]
a (nm)
c (nm)
u (nm)
CuAlO 2
0.2858
1.6958
0.1099
CuGaO 2
0.2980
1.7100
0.1073 ∗
CuInO 2
0.3292
1.7388
0.1056 ∗
Fig. 3.27 Perovskite
structure (BaTiO 3 ), a A
cell with 12-fold
(cuboctahedrally)
configured Ba, b B cell
with octahedrally
configured Ti
(a)
Ba
O
Ti
(b)
Ba 2+
O 2Ti 4+
3 Crystals
3.4.10 Perovskite Structure
The perovskite structure for ABO 3 materials (calcium titanate, CaTiO 3 , space group 62, Pnma)
(Fig. 3.27) is relevant for ferroelectric semiconductors (cf. Sect. 16.3). It is cubic with the Ca (or
Ba, Sr) ions (charge state 2+) on the corners of the cube, the O ions (2−) on the face centers and
the Ti (4+) in the body center. The lattice is simple cubic, the base is Ca at (0,0,0), O at (1/2,1/2,0),
(1/2,0,1/2) and (0,1/2,1/2) and Ti at (1/2,1/2,1/2). The ferroelectric polarization is typically evoked by
a shift of the negatively and positively charged ions relative to each other. LaAlO 3 (lanthanium aluminate) is available as substrate material (space group 226, Fm ¯
3c [227]). Perovskites are also important
for materials for high temperature superconductivity.
Another group of ABX 3 perovskites is formed by halogen atoms (Cl, Br, I) instead of oxygen
on the X-site (in −1 charge state). The B-site (former Ti-position) is taken up by lead (Pb) or other
elements of same +2 valence. The A-site (former Ba-position) is populated with an organic molecule
(+1) such as CH 3 NH 3 (methylammonium, or MA) [228] or HC(NH 2 ) 2 (formamidinium, or FA). In
a purely inorganic version of such halogen perovskite, the A-site site is populated with Cs [229]. In
order to avoid Pb, CsSnX 3 has been investigated. Various phases are observed next to the cubic phase
(α-CsPbI 3 ) such as orthorhombic (δ-CsPbBr 3 ) or tetragonal [230]. Also, angular distortions of the
PbX 6 octahedrons are common and depend on material composition (Fig. 3.28).
The phase stability depends on the octahedral factor μ = R B /R X and the tolerance factor t,
t =
R A + R X
√
2(R B + R X )
,
(3.12)
with R A , R B and R X denoting the radius of the A-site and B-site cation, and X-site anion, respectively. The stability of 138 different halide perovskites has been calculated in terms of μ and t [231]
(Fig. 3.29) and stability criteria have been given. A stable cubic phase requires 0.44 < μ < 0.9 and 0.8
≤ t ≤ 1 [232].
Table 3.5 Lattice parameters a, c, and u of some delafossite compounds. Theoretical values are shown with asterisk.
Data from [226]
a (nm)
c (nm)
u (nm)
CuAlO 2
0.2858
1.6958
0.1099
CuGaO 2
0.2980
1.7100
0.1073 ∗
CuInO 2
0.3292
1.7388
0.1056 ∗
Fig. 3.27 Perovskite
structure (BaTiO 3 ), a A
cell with 12-fold
(cuboctahedrally)
configured Ba, b B cell
with octahedrally
configured Ti
(a)
Ba
O
Ti
(b)
Ba 2+
O 2Ti 4+