386
13 Two-Dimensional Semiconductors
(a)
(b)
Fig. 13.4 a Band structure of graphene from first principles (zero energy refers to the intrinsic Fermi level). The circle
focusses on the linear band crossing at the Fermi level. Adapted from [1247], reprinted with permission, ©2004 WILEYVCH. b Three-dimensional representation E(k x , k y ) of the π-bands of graphene (left) and of graphene-like material
with gap (right)
Fig. 13.5 Band structure of graphene, silicene and germanene as labeled. Adapted from [1255], reprinted with permission, ©2014 RSC
where t ≈ 2.8 eV is the next-neighbor hopping energy (cf. Fig. G.1). More elaborate tight-binding
schemes have been reported [1252, 1365]. The second next-neighbor energy t
is about a factor of
10–100 smaller; a tight-binding fit to cyclotron resonance experiments finds t
= 0.1 eV [1253].
The two-dimensional band structure is visualized in Fig. 13.4b. Such band structure has been directly
confirmed experimentally [1254] as shown in Fig. 13.6a–e. The distortion of the band structure with
increasing electron concentration (Fig. 13.6e–h) from the conical bands is due to strong electronelectron, electron-phonon, and electron-plasmon coupling effects [1254].
The band structures of atomic sheets of C, Si and Ge are compared in Fig. 13.5. The bands cross for
all three materials at the K-point while the band gap at the zone center decreases with increases order
number. Any kind of asymmetry between the A- and B-sites, as might occur for example on certain
corrugated substrates, will introduce a band gap at the Dirac points (Fig. 13.4b).
The band structure of FLG has been theoretically analyzed in [1256, 1257]. For bilayers experimental data on the band structure can be found in [1258]. Subtle differences exist for different stacking
orders of the graphene sheets (see below, Sect. 13.3). Bulk graphite shows a semi-metallic behavior
with a band overlap of about 41 meV. For more than ten graphene layers the difference with the band
overlap in bulk graphite is less than 10%.
13 Two-Dimensional Semiconductors
(a)
(b)
Fig. 13.4 a Band structure of graphene from first principles (zero energy refers to the intrinsic Fermi level). The circle
focusses on the linear band crossing at the Fermi level. Adapted from [1247], reprinted with permission, ©2004 WILEYVCH. b Three-dimensional representation E(k x , k y ) of the π-bands of graphene (left) and of graphene-like material
with gap (right)
Fig. 13.5 Band structure of graphene, silicene and germanene as labeled. Adapted from [1255], reprinted with permission, ©2014 RSC
where t ≈ 2.8 eV is the next-neighbor hopping energy (cf. Fig. G.1). More elaborate tight-binding
schemes have been reported [1252, 1365]. The second next-neighbor energy t
is about a factor of
10–100 smaller; a tight-binding fit to cyclotron resonance experiments finds t
= 0.1 eV [1253].
The two-dimensional band structure is visualized in Fig. 13.4b. Such band structure has been directly
confirmed experimentally [1254] as shown in Fig. 13.6a–e. The distortion of the band structure with
increasing electron concentration (Fig. 13.6e–h) from the conical bands is due to strong electronelectron, electron-phonon, and electron-plasmon coupling effects [1254].
The band structures of atomic sheets of C, Si and Ge are compared in Fig. 13.5. The bands cross for
all three materials at the K-point while the band gap at the zone center decreases with increases order
number. Any kind of asymmetry between the A- and B-sites, as might occur for example on certain
corrugated substrates, will introduce a band gap at the Dirac points (Fig. 13.4b).
The band structure of FLG has been theoretically analyzed in [1256, 1257]. For bilayers experimental data on the band structure can be found in [1258]. Subtle differences exist for different stacking
orders of the graphene sheets (see below, Sect. 13.3). Bulk graphite shows a semi-metallic behavior
with a band overlap of about 41 meV. For more than ten graphene layers the difference with the band
overlap in bulk graphite is less than 10%.