Graphene band structure

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

Fig. 25 Calculated band structure of single-layer graphene. (Reprinted with permission from [193])...

McCann E (2006) Asymmetry gap in the electronic band structure of bilayer graphene. Phys Rev B 74 161403... 

Let us recall that nanotubes can be considered as graphene sheets rolled up in different ways. If we consider the so-called chiral vectors c = nai + na2, in which a and a2 are the basis vectors of a 2D graphite lattice, depending on the value of the integers n and m, one can define three families of tubes armchair tubes (n = m), zig-zag tubes (n or m = 0), and chiral tubes (n m 0). Band structure calculations have demonstrated that tubes are either metallic compounds, or zero-gap semiconductors, or semiconductors [6,7]. More commonly, they are divided into metallic tubes (when n-m is a multiple of 3) or semiconducting ones. 

In two dimensions, these functions are surfaces. Eigure 5.3 shows the band structure diagram of graphene. 

Figure 5.3. The tight-binding band structure for graphene. The electronic properties are well described by the irand tt bands, which intersect atthe point K, making graphene a semi-metal.

Fig. 9.3 Energy band structure of graphene calculated with tight-binding approximation...

Fig. 9.4 Energy band structure of graphene. Solid thick lines density functional theory with local density approximation circles and dashed lines GW approximation [8]...

It is essential to give a correct and conclusive description of the electronic properties of carbon nanotubes to get an understanding of their broad potential for appHcations. Chemists and physicists have developed two fundamentally different concepts of the matter. They are either based on considering electrons in molecular orbitals, especially in frontier orbitals, and on examining the Jt-system, or they pursue a soHd-state physical approach that employs density-of-state functions and the band structure in the Brillouin zone of a two-dimensional graphene. 

This conclusion by analogy shall be explained in the following. Instead of the one-dimensional polyacetylene, a structure completely conjugated in two dimensions, that is, a graphene layer, will be considered. Due to this dimensional extension the calculated electronic band structure can no longer be drawn in dependence on just one parameter (the wavenumber k), but it has to be plotted over the two-dimensional Brillouin zone (Figure 3.49a). The contour lines represent different... 

Figure 3.50 Band structure along a basal plane of graphene (a) ( Wiley-VCH 2004) and representation of the n- and tc -orbitals above the Brillouin zone of graphene (b) ( ACS 2002).

---