3.4 Important Crystal Structures
49
Table 3.4 Lattice nonideality parameters η and u (from (3.10)) of various chalcopyrite compounds and their experimentally observed disorder stability (+/− indicates compound with/without order–disorder (D–O) transition, respectively).
Data from [216]
η
u
D–O
η
u
D–O
CuGaSe 2
0.983
0.264
+
ZnSiAs 2
0.97
0.271
−
CuInSe 2
1.004
0.237
+
ZnGeAs 2
0.983
0.264
+
AgGaSe 2
0.897
0.287
−
CdSiAs 2
0.92
0.294
−
AgInSe 2
0.96
0.261
+
CdGeAs 2
0.943
0.287
CuGaS 2
0.98
0.264
ZnSiP 2
0.967
0.272
−
CuInS 2
1.008
0.236
+
ZnGeP 2
0.98
0.264
+
AgGaS 2
0.895
0.288
−
CdSiP 2
0.92
0.296
−
AgInS 2
0.955
0.262
CdGeP 2
0.939
0.288
−
along [100] and [010] such that it is closer (if d AC < d BC ) to the pair of A atoms and further away from
the B atoms. The displacement parameter is
u =
1
4
+
R
2
AC − R
2
BC
a 2
(3.8)
and it deviates from the ideal value u 0 = 1/4 for the zincblende structure as listed in Table 3.4 for a
number of chalcopyrite compounds. In the chalcopyrite structure
R AC = a
u 2 +
1 + η 2
16
(3.9a)
R BC = a
u −
1
2
2
+
1 + η 2
16
.
(3.9b)
The minimization of the microscopic strain yields (in first order) [217]
u ∼ =
1
4
+
3
8
d
2
AC − d
2
BC
d
2
AC + d
2
BC
.
(3.10)
Compounds with u > u c , u c = 0.265 being a critical displacement parameter, (or u < 1/2 − u c =
0.235) are stable with regard to cation disorder [216]. In Fig. 3.23 the correlation of the calculated
value for u according to (3.10) and the experimental values is shown.
3.4.7 Spinel Structure
A large variety of ternary compounds of type A
II B
III
2 C
VI
4 crystallize inb the cubic spinel structure
(spinel, MgAl 2 O 4 , space group 227, Fd3m). Typical elements are A: Mg, Cr, Mn, Fe, Co, Ni, Cu, Zn,
Cd, Sn, B: Al, Ga, In, Ti, V, Cr, Mn, Fe, Co, Rh, Ni, and C: O, S, Se, Te.
As an example ZnGa 2 O 4 (zinc gallate) has received attention as interfacial layer in ZnO/GaAs epitaxy [218], luminescent material [219], and as ferromagnetic semiconductor [220]. Also the (unwanted)
appearance of zinc gallate inclusions is in competition with the formation of highly doped wurtzite
ZnO:Ga; in the (normal) spinel cubic structure of ZnGa 2 O 4 the Zn
2+ ions still occupy tetrahedral
49
Table 3.4 Lattice nonideality parameters η and u (from (3.10)) of various chalcopyrite compounds and their experimentally observed disorder stability (+/− indicates compound with/without order–disorder (D–O) transition, respectively).
Data from [216]
η
u
D–O
η
u
D–O
CuGaSe 2
0.983
0.264
+
ZnSiAs 2
0.97
0.271
−
CuInSe 2
1.004
0.237
+
ZnGeAs 2
0.983
0.264
+
AgGaSe 2
0.897
0.287
−
CdSiAs 2
0.92
0.294
−
AgInSe 2
0.96
0.261
+
CdGeAs 2
0.943
0.287
CuGaS 2
0.98
0.264
ZnSiP 2
0.967
0.272
−
CuInS 2
1.008
0.236
+
ZnGeP 2
0.98
0.264
+
AgGaS 2
0.895
0.288
−
CdSiP 2
0.92
0.296
−
AgInS 2
0.955
0.262
CdGeP 2
0.939
0.288
−
along [100] and [010] such that it is closer (if d AC < d BC ) to the pair of A atoms and further away from
the B atoms. The displacement parameter is
u =
1
4
+
R
2
AC − R
2
BC
a 2
(3.8)
and it deviates from the ideal value u 0 = 1/4 for the zincblende structure as listed in Table 3.4 for a
number of chalcopyrite compounds. In the chalcopyrite structure
R AC = a
u 2 +
1 + η 2
16
(3.9a)
R BC = a
u −
1
2
2
+
1 + η 2
16
.
(3.9b)
The minimization of the microscopic strain yields (in first order) [217]
u ∼ =
1
4
+
3
8
d
2
AC − d
2
BC
d
2
AC + d
2
BC
.
(3.10)
Compounds with u > u c , u c = 0.265 being a critical displacement parameter, (or u < 1/2 − u c =
0.235) are stable with regard to cation disorder [216]. In Fig. 3.23 the correlation of the calculated
value for u according to (3.10) and the experimental values is shown.
3.4.7 Spinel Structure
A large variety of ternary compounds of type A
II B
III
2 C
VI
4 crystallize inb the cubic spinel structure
(spinel, MgAl 2 O 4 , space group 227, Fd3m). Typical elements are A: Mg, Cr, Mn, Fe, Co, Ni, Cu, Zn,
Cd, Sn, B: Al, Ga, In, Ti, V, Cr, Mn, Fe, Co, Rh, Ni, and C: O, S, Se, Te.
As an example ZnGa 2 O 4 (zinc gallate) has received attention as interfacial layer in ZnO/GaAs epitaxy [218], luminescent material [219], and as ferromagnetic semiconductor [220]. Also the (unwanted)
appearance of zinc gallate inclusions is in competition with the formation of highly doped wurtzite
ZnO:Ga; in the (normal) spinel cubic structure of ZnGa 2 O 4 the Zn
2+ ions still occupy tetrahedral