384
13 Two-Dimensional Semiconductors
Fig. 13.1 Unit cell of A–B compound honeycomb lattice in a real space and b reciprocal space. The irreducible part of
the Brillouin zone is shown in dark grey
(a)
(b)
Fig. 13.2 a Schematic layer structure of graphite with bond length and layer distance labelled. b AFM image of graphene
on oxidized silicon. The height of two areas relative to the background is labelled. Adapted from [1240]
13.1.1 Structural Properties
The carbon bond length in graphene is d C−C = 0.142 nm (cmp. Table 13.1). Due to instabilities
[1234, 1235], freely suspended graphene layers are not perfectly flat but exhibit corrugations [1236].
Graphene can be prepared from graphite via micromechanical cleavage, i.e. mechanical exfoliation
(repeated peeling) of small mesas of highly oriented pyrolytic graphite [1237, 1238]. Epitaxial growth
on SiC [1239] and the growth on copper in large sheets [1232] have been reported.
Graphite is a stacked arrangement of such graphene sheets (Fig. 13.2a) held together by van-derWaals forces (cmp. Sect. 13.3) as shown in Fig. 13.2b. The carbon atoms bond via sp
2 hybridization.
Organic molecules such as, e.g., anthracene or coronene (Fig. 18.1) can be understood as molecular-size
pieces of such two-dimensional graphene sheet with hydrogen saturating the outside, broken bonds.
Single layer graphene sheets (SLG) and few-layer graphene (FLG) sheets must distinguished since for
the latter the exact stacking arrangement and vertical coupling effects play a role. Ideally, such twodimensional crystal is infinitely extended, e.g. for band structure calculations. Of course real crystals
always have a (quasi-) one-dimensional boundary which is topologically a line or very thin sidewall.
Various defects have been studied in graphene. The vacancy is discussed in [1241] and exhibits a
magnetic moment of 2μ B .
The unit cells in real and reciprocal space are shown in Fig. 13.1. The lattice vectors are a 1,2 =
a/2(3, ±
√
3) (Fig. 13.1a). The reciprocal lattice vectors are b 1,2 = 2π/(3a)(1, ±
√
3), where a =
d C−C . The Brillouin zone (Fig. 13.1b) has two triples of K-points that are labeled K and K’ (or K+
13 Two-Dimensional Semiconductors
Fig. 13.1 Unit cell of A–B compound honeycomb lattice in a real space and b reciprocal space. The irreducible part of
the Brillouin zone is shown in dark grey
(a)
(b)
Fig. 13.2 a Schematic layer structure of graphite with bond length and layer distance labelled. b AFM image of graphene
on oxidized silicon. The height of two areas relative to the background is labelled. Adapted from [1240]
13.1.1 Structural Properties
The carbon bond length in graphene is d C−C = 0.142 nm (cmp. Table 13.1). Due to instabilities
[1234, 1235], freely suspended graphene layers are not perfectly flat but exhibit corrugations [1236].
Graphene can be prepared from graphite via micromechanical cleavage, i.e. mechanical exfoliation
(repeated peeling) of small mesas of highly oriented pyrolytic graphite [1237, 1238]. Epitaxial growth
on SiC [1239] and the growth on copper in large sheets [1232] have been reported.
Graphite is a stacked arrangement of such graphene sheets (Fig. 13.2a) held together by van-derWaals forces (cmp. Sect. 13.3) as shown in Fig. 13.2b. The carbon atoms bond via sp
2 hybridization.
Organic molecules such as, e.g., anthracene or coronene (Fig. 18.1) can be understood as molecular-size
pieces of such two-dimensional graphene sheet with hydrogen saturating the outside, broken bonds.
Single layer graphene sheets (SLG) and few-layer graphene (FLG) sheets must distinguished since for
the latter the exact stacking arrangement and vertical coupling effects play a role. Ideally, such twodimensional crystal is infinitely extended, e.g. for band structure calculations. Of course real crystals
always have a (quasi-) one-dimensional boundary which is topologically a line or very thin sidewall.
Various defects have been studied in graphene. The vacancy is discussed in [1241] and exhibits a
magnetic moment of 2μ B .
The unit cells in real and reciprocal space are shown in Fig. 13.1. The lattice vectors are a 1,2 =
a/2(3, ±
√
3) (Fig. 13.1a). The reciprocal lattice vectors are b 1,2 = 2π/(3a)(1, ±
√
3), where a =
d C−C . The Brillouin zone (Fig. 13.1b) has two triples of K-points that are labeled K and K’ (or K+