3.3 Lattice
37
Fig. 3.3 The
two-dimensional Bravais
lattices with the primitive
unit cells: a square lattice
(a = b, φ = 90 ◦ ), b
hexagonal lattice (a = b,
φ = 60 ◦ ), c rectangular
lattice (a = b, φ = 90 ◦ ), d
centered-rectangular lattice
(a = b, φ = 90 ◦ , for the
(nonprimitive) rectangular
unit cell shown on the
right)
a
b
a
b
a
b
a
b
a
b
(a)
(b)
(c)
(d)
3.3.2 3D Bravais Lattices
In three dimensions, the operations of the point group results in fourteen 3D Bravais lattices (Fig. 3.4),
that are categorized into seven crystal classes (trigonal, monoclinic, rhombic, tetragonal, cubic, rhombohedral and hexagonal). These classes are discerned by the conditions for the lengths and the mutual
angles of the vectors that span the lattice (Table 3.1). Some classes have several members. The cubic
crystal can have a simple (sc), face-centered (fcc) or body-centered (bcc) lattice.
In the following, some of the most important lattices, in particular those most relevant to semiconductors, will be treated in some more detail.
3.3.2.1 Cubic fcc and bcc Lattices
The primitive translation vectors for the cubic face-centered (fcc) and the cubic body-centered (bcc)
lattice are shown in Fig. 3.5 and Fig. 3.6, respectively. Many metals crystallize in these lattices, e.g.
copper (fcc) and tungsten (bcc).
Table 3.1 Conditions for lengths and angles for the 7 crystal classes. Note that only the positive conditions are listed.
The rhombohedral system is a special case of the trigonal class. Conditions for the trigonal and hexagonal classes are
the same, however, trigonal symmetry includes a single C 3 or S 6 axis, while hexagonal symmetry includes a single C 6
or S 5
6 axis
System
#
Lattice
Conditions for the
symbol
usual unit cell
Triclinic
1
None
Monoclinic
2
s, c
α = γ = 90 ◦ or
α = β = 90 ◦
Orthorhombic
4
s, c, bc, fc
α = β = γ = 90 ◦
Tetragonal
2
s, bc
a = b , α = β = γ = 90 ◦
Cubic
3
s, bc, fc
a = b = c ,
α = β = γ = 90 ◦
Trigonal
1
a = b , α = β = 90 ◦ ,
γ = 120 ◦
(Rhombohedral)
1
a = b = c , α = β = γ
Hexagonal
1
a = b , α = β = 90 ◦ ,
γ = 120 ◦
37
Fig. 3.3 The
two-dimensional Bravais
lattices with the primitive
unit cells: a square lattice
(a = b, φ = 90 ◦ ), b
hexagonal lattice (a = b,
φ = 60 ◦ ), c rectangular
lattice (a = b, φ = 90 ◦ ), d
centered-rectangular lattice
(a = b, φ = 90 ◦ , for the
(nonprimitive) rectangular
unit cell shown on the
right)
a
b
a
b
a
b
a
b
a
b
(a)
(b)
(c)
(d)
3.3.2 3D Bravais Lattices
In three dimensions, the operations of the point group results in fourteen 3D Bravais lattices (Fig. 3.4),
that are categorized into seven crystal classes (trigonal, monoclinic, rhombic, tetragonal, cubic, rhombohedral and hexagonal). These classes are discerned by the conditions for the lengths and the mutual
angles of the vectors that span the lattice (Table 3.1). Some classes have several members. The cubic
crystal can have a simple (sc), face-centered (fcc) or body-centered (bcc) lattice.
In the following, some of the most important lattices, in particular those most relevant to semiconductors, will be treated in some more detail.
3.3.2.1 Cubic fcc and bcc Lattices
The primitive translation vectors for the cubic face-centered (fcc) and the cubic body-centered (bcc)
lattice are shown in Fig. 3.5 and Fig. 3.6, respectively. Many metals crystallize in these lattices, e.g.
copper (fcc) and tungsten (bcc).
Table 3.1 Conditions for lengths and angles for the 7 crystal classes. Note that only the positive conditions are listed.
The rhombohedral system is a special case of the trigonal class. Conditions for the trigonal and hexagonal classes are
the same, however, trigonal symmetry includes a single C 3 or S 6 axis, while hexagonal symmetry includes a single C 6
or S 5
6 axis
System
#
Lattice
Conditions for the
symbol
usual unit cell
Triclinic
1
None
Monoclinic
2
s, c
α = γ = 90 ◦ or
α = β = 90 ◦
Orthorhombic
4
s, c, bc, fc
α = β = γ = 90 ◦
Tetragonal
2
s, bc
a = b , α = β = γ = 90 ◦
Cubic
3
s, bc, fc
a = b = c ,
α = β = γ = 90 ◦
Trigonal
1
a = b , α = β = 90 ◦ ,
γ = 120 ◦
(Rhombohedral)
1
a = b = c , α = β = γ
Hexagonal
1
a = b , α = β = 90 ◦ ,
γ = 120 ◦