60
3 Crystals
Table 3.6 High symmetry points and directions from -point in the Brillouin zone of the fcc lattice
Point
k (
2 π
a )
Direction
Multiplicity
(0, 0, 0)
1
X
(0, 1, 0)
6
K
3/4 (1, 1, 0)
12
L
1/2 (1, 1, 1)
8
W
(1, 1/2, 0)
24
U
(1, 1/4, 1/4)
24
Table 3.7 High symmetry points and directions from -point in the Brillouin zone of the hcp lattice
Point
k (2 π )
Direction
Multiplicity
(0, 0, 0)
1
A
(0, 0,
1
2 c )
2
L
(0,
1
√
3 a
,
1
2 c )
12
M
(0,
1
√
3 a
, 0)
6
H
(−
1
3 a ,
1
√
3 a
,
1
2 c )
12
K
(−
1
3 a ,
1
√
3 a
, 0)
T
6
cos θ
t
=
l
l 2 +
c 2
a 2 (h 2 + k 2 )
(3.22b)
cos θ
h
=
l
l 2 +
4
3
c 2
a 2 (h 2 + h k + k 2 )
.
(3.22c)
3.6.3 Brillouin Zone
The Wigner–Seitz cell in reciprocal space is called the (first) Brillouin zone. In Fig. 3.38, the Brillouin
zones for the most important lattices are shown. High symmetry points in the Brillouin zone are labeled
with dedicated letters. The point always denotes k = 0 (zone center). High symmetry paths in the
Brillouin zone are labeled with dedicated Greek symbols.
In the Brillouin zone of the fcc lattice (Si, Ge, GaAs, ...) the X point denotes the point at the zone
boundary in 001-directions (at distance 2 π/a from ), K for 110-directions (at distance 3 π/
√
2 a
from ) and L for the 111-directions (at distance
√
3 π/a from ) (see Table 3.6). The straight paths
from to X, K, and L are denoted as , , and , respectively. High symmetry points and directions
of the Brillouin zone of the hcp lattice are given in Table 3.7.
3.7 Alloys
When different semiconductors are mixed various cases can occur:
• The semiconductors are not miscible and have a so-called miscibility gap. They will tend to form
clusters that build up the crystal. The formation of defects is probable.