50
3 Crystals
Fig. 3.23 Experimental
values u exp of the
displacement parameter for
various chalcopyrites vs.
the calculated value u calc
according to (3.10). The
dashed line indicates
u exp = u calc . Adapted
from [217]
Fig. 3.24 Spinel AB 2 C 4
crystal structure, the
cations are depicted as
yellow (A atoms) and silver
(B atoms) spheres, the
anions (C atoms) as blue
spheres
sites, but the Ga
3+ ions occupy octahedral sites instead of tetrahedral sites in doped wurtzite ZnO:Ga.
(Sc,Al)MgO 4 (SCAM) is available as substrate material. Also A
VI B
II
2 C
VI
4 compounds exist, e.g.
GeB 2 O 4 (with B=Mg, Fe, Co, Ni).
The anion atoms (C
2− ) sit on a fcc lattice. The A atoms fill 1/8 of all tetraeder spaces and the B
atoms fill half of all octaeder places (Fig. 3.24). Often the cations are charged A
2+ and B
3+ , e.g. in
ZnAl 2 O 4 , MgCr 2 O 4 or ZnCo 2 O 4 . Also A
6+ and B
1− exists, e.g. in WNa 2 O 4 .
The cubic lattice constant is denoted as a. In real spinels, the anions deviate from the ideal fcc array
which is accounted for by the parameter u, measuring the displacement of anions along the [111]direction [221]; if the A-site cation is at (0,0,0), an anion is at (u, u, u). The cation-anion distances are
given by [222]
R AC = a
3 (u − 1/8) ,
(3.11a)
R BC = a
3 u 2 − 2 u − 3/8 .
(3.11b)
The ideal value is u = 1/4; examples are u = 0.2624 for MgAl 2 O 4 , u = 0.2617 for ZnGa 2 O 4 and
u = 0.2409 for SiFe 2 O 4 [222].
In the inverted spinel structure, for A
II B
III
2 C
VI
4 compounds, the cations are distributed like
B(AB)C 4 , i.e. the B cations occupy tetraeder and octaeder places, e.g. in Mg
2+ (Mg
2+ Ti
4+ )O
2−
4 or
Fe
3+ (Ni
2+ Fe
3+ )O
2−
4 . Examples are magnetite (Fe 3 O 4 ), a material with high spin polarization, or
MgFe 2 O 4 . Also A
VI B
II
2 C
VI
4 compounds exist in this structure, e.g. SnB 2 O 4 (with B = Mg, Mn,
Co, Zn), TiB 2 O 4 (with B = Mg, Mn, Fe, Co, Zn), and VB 2 O 4 (with B = Mg, Co, Zn).
3 Crystals
Fig. 3.23 Experimental
values u exp of the
displacement parameter for
various chalcopyrites vs.
the calculated value u calc
according to (3.10). The
dashed line indicates
u exp = u calc . Adapted
from [217]
Fig. 3.24 Spinel AB 2 C 4
crystal structure, the
cations are depicted as
yellow (A atoms) and silver
(B atoms) spheres, the
anions (C atoms) as blue
spheres
sites, but the Ga
3+ ions occupy octahedral sites instead of tetrahedral sites in doped wurtzite ZnO:Ga.
(Sc,Al)MgO 4 (SCAM) is available as substrate material. Also A
VI B
II
2 C
VI
4 compounds exist, e.g.
GeB 2 O 4 (with B=Mg, Fe, Co, Ni).
The anion atoms (C
2− ) sit on a fcc lattice. The A atoms fill 1/8 of all tetraeder spaces and the B
atoms fill half of all octaeder places (Fig. 3.24). Often the cations are charged A
2+ and B
3+ , e.g. in
ZnAl 2 O 4 , MgCr 2 O 4 or ZnCo 2 O 4 . Also A
6+ and B
1− exists, e.g. in WNa 2 O 4 .
The cubic lattice constant is denoted as a. In real spinels, the anions deviate from the ideal fcc array
which is accounted for by the parameter u, measuring the displacement of anions along the [111]direction [221]; if the A-site cation is at (0,0,0), an anion is at (u, u, u). The cation-anion distances are
given by [222]
R AC = a
3 (u − 1/8) ,
(3.11a)
R BC = a
3 u 2 − 2 u − 3/8 .
(3.11b)
The ideal value is u = 1/4; examples are u = 0.2624 for MgAl 2 O 4 , u = 0.2617 for ZnGa 2 O 4 and
u = 0.2409 for SiFe 2 O 4 [222].
In the inverted spinel structure, for A
II B
III
2 C
VI
4 compounds, the cations are distributed like
B(AB)C 4 , i.e. the B cations occupy tetraeder and octaeder places, e.g. in Mg
2+ (Mg
2+ Ti
4+ )O
2−
4 or
Fe
3+ (Ni
2+ Fe
3+ )O
2−
4 . Examples are magnetite (Fe 3 O 4 ), a material with high spin polarization, or
MgFe 2 O 4 . Also A
VI B
II
2 C
VI
4 compounds exist in this structure, e.g. SnB 2 O 4 (with B = Mg, Mn,
Co, Zn), TiB 2 O 4 (with B = Mg, Mn, Fe, Co, Zn), and VB 2 O 4 (with B = Mg, Co, Zn).