13.1 Graphene and Related Materials
385
(a)
(b)
Fig. 13.3 a Phonon dispersion in graphene (cmp. Fig. 13.1b). Symbols are experimental data from various methods.
Dashed lines is DFT-LDA theory, solid lines GGA. Adapted from [1244], reprinted with permission, ©2004 Elsevier Ltd.
b Comparison of graphite and graphene Raman spectra (excited at 514 nm). In the right panel the number of monolayers
in FLG is labeled (1: graphene, ∞: graphite). Adapted from [1245], reprinted with permission, ©2006 APS
and K−) at K , K
= 2π/(3a) (1, ±1/
√
3). When the A- and B-sites are identical, as for graphene, the
difference between K- and K’-points is often subtle and a –M–K Brillouin zone is sufficient.
Mechanical properties of graphene are discussed in [1242, 1243]; it has exceptional strength
(Young’s modulus in the range of 1 TPa). The breaking stress 200 times greater than steel, with a
2D tensile stress of 42 N/m, corresponding to a strain of 25%. The phonon dispersion of graphene is
shown in Fig. 13.3a. For the ZA and ZO modes, the displacement is perpendicular to the graphene plane
(out-of plane modes). In Fig. 13.3b the Raman spectra of graphite and graphene are compared. The
peak (termed ‘G’) at about 1580 cm
−1 ist due to the zone center mode. The peak at about 2680 cm
−1
(termed ‘2D’) is due to two phonons (double resonance) with opposite momentum in the highest optical
branch near the K-point. It splits into four peaks for bilayer graphene due to changes in the electronic
band structure [1245].
13.1.2 Band Structures
The band structure of graphene from first principles [1246] is depicted in Fig. 13.4a. Graphene is a
zero-gap semiconductor (cf. Fig. 6.46) which shows a linear photon-like spectrum,
E = k v F ,
(13.1)
around the Fermi energy at the K-point (Fig. 13.4b). Such point is also called ’Dirac’ point.
The important bands close to the Fermi level stem from the (out-of-plane) π-orbitals while the (inplane) chemical bonds are sp
2 -type (Sect. 2.2.3). The linear dispersion around the K-point is similar
to that of relativistic particles without rest mass. The electrons in graphene are of course not really
massless, their velocity (6.35) being v F ≈ 10
6 m/s, about 300 times smaller than the speed of light [1248,
1249].
In the simplest tight-binding approximation (cf. Appendix G), considering the coupling of the carbon
(out-of-plane) p z -orbitals with three nearest neighbors, the band structure is given as [1250, 1251] (cf.
(G.23)
E(k) = ±t
1 + 4 cos
3
a k x
2
cos
√
3 a k y
2
+ 4 cos 2
√
3 a k y
2
,
(13.2)
385
(a)
(b)
Fig. 13.3 a Phonon dispersion in graphene (cmp. Fig. 13.1b). Symbols are experimental data from various methods.
Dashed lines is DFT-LDA theory, solid lines GGA. Adapted from [1244], reprinted with permission, ©2004 Elsevier Ltd.
b Comparison of graphite and graphene Raman spectra (excited at 514 nm). In the right panel the number of monolayers
in FLG is labeled (1: graphene, ∞: graphite). Adapted from [1245], reprinted with permission, ©2006 APS
and K−) at K , K
= 2π/(3a) (1, ±1/
√
3). When the A- and B-sites are identical, as for graphene, the
difference between K- and K’-points is often subtle and a –M–K Brillouin zone is sufficient.
Mechanical properties of graphene are discussed in [1242, 1243]; it has exceptional strength
(Young’s modulus in the range of 1 TPa). The breaking stress 200 times greater than steel, with a
2D tensile stress of 42 N/m, corresponding to a strain of 25%. The phonon dispersion of graphene is
shown in Fig. 13.3a. For the ZA and ZO modes, the displacement is perpendicular to the graphene plane
(out-of plane modes). In Fig. 13.3b the Raman spectra of graphite and graphene are compared. The
peak (termed ‘G’) at about 1580 cm
−1 ist due to the zone center mode. The peak at about 2680 cm
−1
(termed ‘2D’) is due to two phonons (double resonance) with opposite momentum in the highest optical
branch near the K-point. It splits into four peaks for bilayer graphene due to changes in the electronic
band structure [1245].
13.1.2 Band Structures
The band structure of graphene from first principles [1246] is depicted in Fig. 13.4a. Graphene is a
zero-gap semiconductor (cf. Fig. 6.46) which shows a linear photon-like spectrum,
E = k v F ,
(13.1)
around the Fermi energy at the K-point (Fig. 13.4b). Such point is also called ’Dirac’ point.
The important bands close to the Fermi level stem from the (out-of-plane) π-orbitals while the (inplane) chemical bonds are sp
2 -type (Sect. 2.2.3). The linear dispersion around the K-point is similar
to that of relativistic particles without rest mass. The electrons in graphene are of course not really
massless, their velocity (6.35) being v F ≈ 10
6 m/s, about 300 times smaller than the speed of light [1248,
1249].
In the simplest tight-binding approximation (cf. Appendix G), considering the coupling of the carbon
(out-of-plane) p z -orbitals with three nearest neighbors, the band structure is given as [1250, 1251] (cf.
(G.23)
E(k) = ±t
1 + 4 cos
3
a k x
2
cos
√
3 a k y
2
+ 4 cos 2
√
3 a k y
2
,
(13.2)