3.6 Reciprocal Lattice
59
Fig. 3.36 Sketch of lattice
directions and planes in a
monoclinic lattice. The
[100] direction and the
normal of the (100) plane
are not parallel
Fig. 3.37 (a, b) Miller
indices for the wurtzite (or
hcp) structure. (c)
Orientation of the a-, r-,
m-, and c-plane in the
wurtzite structure
(a)
(b)
[1120]
(00.1)
[1120]
[2110]
[1210]
[1210]
[2110]
[1010]
[0110]
[0110]
[1010]
[1100]
[1100]
(c)
a
r
m
c
In the wurtzite lattice, the Miller indices are denoted as [hklm] (Fig. 3.37). Within the (0001) plane
three indices hkl are used that are related to the three vectors a 1 , a 2 and a 3 (see Fig. 3.37a) rotated with
respect to each other by 120
◦ . Of course, the four indices are not independent and l = −(h + k). The
third (redundant) index can be denoted as a dot. The c-axis [0001] is then denoted as [00.1]. Wurtzite
(and trigonal, e.g. sapphire) substrates are available typically with (polished) a (11.0), m (01.0) and
r (01.2), c (00.1) surfaces (Fig. 3.37b).
The distance of lattice planes d = 2π/|G| can be expressed via the Miller indices for cubic (3.21a),
tetragonal (3.21b) and hexagonal (3.21c) crystals as
d
c
hkl =
a
√
h 2 + k 2 + l 2
(3.21a)
d
t
hkl =
a
h 2 + k 2 + l 2 (a/c) 2
(3.21b)
d
h
hkl =
a
4 (h 2 + hk + k 2 )/3 + l 2 (a/c) 2
(3.21c)
Useful formulas for the angle θ between a [hk.l]-plane and the [00.1]-direction in the cubic, tetragonal
and wurtzite structures are:
cos θ
c
=
l
√
h 2 + k 2 + l 2
(3.22a)