340
11 Surfaces
tronic properties (surface states) are different from the bulk modes. Detailed treatments of surface
physics and its experimental methods can be found in [695, 1056–1058].
11.2 Surface Crystallography
The surface symmetry, i.e. the two-dimensional spatial periodicity of the constituent atoms, is described
with the ten two-dimensional point groups (Table B.1). The point symmetries are 1-, 2-, 3-, 4-, and 6-fold
rotational symmetry (Fig. 3.9) with or without mirror plane(s). The two-dimensional point symmetries
of popular substrate orientations are compiled in Table 11.1; the different symmetry of the first layer
and the half-space are notable. The combination of five 2D Bravais lattices (cmp. Sect. 3.3.1) with the
10 2D point groups leads to the 17 two-dimensional space groups (wallpaper groups) [1059].
For the treatment of surface in reciprocal space, the three-dimensional k-vector is split into the
two-dimensional component k || parallel and the one-dimensional component k ⊥ perpendicular to the
surface,
k = k || + k ⊥ .
(11.1)
The three most important Brillouin zones in two-dimensional k-space are depicted in Fig. 11.1. Often
the special points of the 2D Brillouin zones are denoted with a bar over the letter.
Table 11.1 Two-dimensional point symmetries of common substrates with ideal low-index surfaces
Crystal
Surface
1st layer
1st & 2nd layers
Half space
Rocksalt
(001)
4 mm
4 mm
4 mm
(110)
2 mm
2 mm
2 mm
(111)
6 mm
3 m
3 m
Diamond
(001)
4 mm
2 mm
2 mm
(110)
2 mm
2 mm
2 mm
(111)
6 mm
3 m
3 m
Zincblende
(001)
4 mm
2 mm
2 mm
(110)
1 m
1 m
1 m
(111)
6 mm
3 m
3 m
Wurtzite
(00.1)
6 mm
3 m
3 m
(10.1)
2 mm
2 mm
1 m
(11.0)
2 mm
2 mm
1 m
M
J'
J
(a)
K M
(c)
'
X
M
X
(b)
Fig. 11.1 Two-dimensional Brillouin zones for a square, b rectangular and c hexagonal surface symmetry. Special
points are labelled and the grey areas indicate the smallest irreducible area