58
3 Crystals
Fig. 3.34 The plane
intersects the axes at 3, 2,
and 2. The inverse of these
numbers is 1/3, 1/2, and
1/2. The smallest integer
numbers of this ratio form
the Miller indices (233)
Fig. 3.35 a Miller indices
of important planes for the
simple cubic (and fcc, bcc)
lattice. (b) Directions
within three low index
planes of cubic crystals
(a)
(b)
[001]
[111]
[110]
(110)
[110]
[001]
[112]
[010] [110]
[100]
(001)
[001] [110]
[100]
[110]
[110]
[112]
[211]
[110]
[121]
(111)
[110]
[112]
[211]
[121]
[111]
[111]
[111]
In a cubic lattice, the faces of the cubic unit cell are {001} and the planes perpendicular to the area
(body) diagonals are {110} ({111}) (Fig. 3.35a). For example, in the simple cubic lattice (100), (010),
(001), (−1 00), (0−1 0) are (00−1 ) equivalent and are denoted by {100}.
In the zincblende lattice, the {111} planes consist of diatomic planes with Zn and S atoms. It depends
on the direction whether the metal or the nonmetal is on top. These two cases are denoted by A and B.
We follow the convention that the (111) plane is (111)A and the metal is on top (as in Fig. 3.16b). For
each change of sign the type changes from A to B and vice versa, e.g. (111)A, (1 ¯
1 1)B and ( ¯
1 ¯
1 ¯
1)B. In
Fig. 3.35b the in-plane directions for the (001), (110) and (111) planes are visualized.
We note that for orthogonal lattices, the (outward) normal direction of a plane (hkl) is the same
direction as [hkl]. Caveat: in non-rectangular lattices this is not the case! This can be easily seen for a
monoclinic lattice, depicted as example in Fig. 3.36.