54
3 Crystals
Fig. 3.30 NiAs structure,
metal atoms: dark grey,
chalcogenide atoms: light
grey
(a)
(b)
• corundum structure (Al 2 O 3 , space group 167, R ¯
3c) occurring, e.g., for sapphire substrates used in
epitaxy or for gallium oxide α-Ga 2 O 3 (Ga 2 O 3 is a multiphase material [234, 235])
• bixbyite structure (In 2 O 3 , δ-Ga 2 O 3 , space group 206, Ia ¯
3) (see Fig. 20.3)
• β-Ga 2 O 3 monoclinic structure (space group 12, C2/m) [235]
• quartz (SiO 2 ) structures, α-quartz (space group 154, P3 2 21) and β-quartz (space group 180, P6 2 22)
Space does not permit to discuss these and other structures in more detail here. The reader should
refer to textbooks on crystallography, e.g. [236–238], and space groups [195, 239]. A good source for
information and images of crystal structures on the web is [240].
3.5 Polytypism and Polymorphism
In polytype materials, several stacking orders are possible. One of them represents the thermodynamic
ground state but others have very similar energy. An example is GaN which has been extensively
investigated in its (equilibrium) wurtzite and also in its (zincblende) cubic form. But not only hcp
or fcc but many different sequences, such as, e.g., ACBCABAC as the smallest unit cell along the
stacking direction, are possible. A typical example is SiC, for which besides hcp and fcc many other
(>40) stacking sequences are known. The largest primitive unit cell of SiC [213] contains 594 layers.
Some of the smaller polytypes are sketched in Fig. 3.31. In Fig.3.32 cubic diamond crystallites and
metastable hexagonal and orthorhombic phases (in silicon) are shown.
For the ternary alloy (cf. Sect. 3.7) Zn 1−x Cd x S the numbers n h of diatomic layers with hexagonal
stacking (AB) and n c of layers with cubic stacking (ABC) have been investigated. CdS has wurtzite
structure and ZnS mostly zincblende structure. The hexagonality index α as defined in (3.13) is shown
in Fig. 3.33 for Zn 1−x Cd x S
α =
n h
n h + n c
.
(3.13)
Some semiconductor materials possess several crystal phases (polymorphs) that can be readily prepared
under various (non-equilibrium) conditions. Due to their different physical properties, eventually they
can be employed for different device applications. GaN with its hexagonal and cubic phases can be
counted as such material. A prominent example of a multi-phase semiconductor is Ga 2 O 3 that exhibits,
next to its monoclinic equilibrium structure, various other phases such as a rhombohedral (corundum,
α-Ga 2 O 3 ) and an orthorhombic (κ-Ga 2 O 3 ) phase [244].
3 Crystals
Fig. 3.30 NiAs structure,
metal atoms: dark grey,
chalcogenide atoms: light
grey
(a)
(b)
• corundum structure (Al 2 O 3 , space group 167, R ¯
3c) occurring, e.g., for sapphire substrates used in
epitaxy or for gallium oxide α-Ga 2 O 3 (Ga 2 O 3 is a multiphase material [234, 235])
• bixbyite structure (In 2 O 3 , δ-Ga 2 O 3 , space group 206, Ia ¯
3) (see Fig. 20.3)
• β-Ga 2 O 3 monoclinic structure (space group 12, C2/m) [235]
• quartz (SiO 2 ) structures, α-quartz (space group 154, P3 2 21) and β-quartz (space group 180, P6 2 22)
Space does not permit to discuss these and other structures in more detail here. The reader should
refer to textbooks on crystallography, e.g. [236–238], and space groups [195, 239]. A good source for
information and images of crystal structures on the web is [240].
3.5 Polytypism and Polymorphism
In polytype materials, several stacking orders are possible. One of them represents the thermodynamic
ground state but others have very similar energy. An example is GaN which has been extensively
investigated in its (equilibrium) wurtzite and also in its (zincblende) cubic form. But not only hcp
or fcc but many different sequences, such as, e.g., ACBCABAC as the smallest unit cell along the
stacking direction, are possible. A typical example is SiC, for which besides hcp and fcc many other
(>40) stacking sequences are known. The largest primitive unit cell of SiC [213] contains 594 layers.
Some of the smaller polytypes are sketched in Fig. 3.31. In Fig.3.32 cubic diamond crystallites and
metastable hexagonal and orthorhombic phases (in silicon) are shown.
For the ternary alloy (cf. Sect. 3.7) Zn 1−x Cd x S the numbers n h of diatomic layers with hexagonal
stacking (AB) and n c of layers with cubic stacking (ABC) have been investigated. CdS has wurtzite
structure and ZnS mostly zincblende structure. The hexagonality index α as defined in (3.13) is shown
in Fig. 3.33 for Zn 1−x Cd x S
α =
n h
n h + n c
.
(3.13)
Some semiconductor materials possess several crystal phases (polymorphs) that can be readily prepared
under various (non-equilibrium) conditions. Due to their different physical properties, eventually they
can be employed for different device applications. GaN with its hexagonal and cubic phases can be
counted as such material. A prominent example of a multi-phase semiconductor is Ga 2 O 3 that exhibits,
next to its monoclinic equilibrium structure, various other phases such as a rhombohedral (corundum,
α-Ga 2 O 3 ) and an orthorhombic (κ-Ga 2 O 3 ) phase [244].