Dirac cones

Fig. 7.3 (a) 2D Brillouin zone of graphene showing characteristic points K and T and Dirac cones located at the six comers (K points), (b) Second-order double resonance scheme for the D peak (close to F) (c) Raman spectral process for the D peak (involving two neighboring K points of the Brillouin zone K and K ). El is the incident laser energy. (After Ref. [46, 48])... 

The extraordinary electronic properties of graphene have spurred the search for other two-dimensional carbon allotropes. Graphene s electronic properties are related to its exhibiting Dirac cones and points, where the valence and conduction bands meet at the Fermi level at these points it may be considered a semiconductor with a zero band gap. The allotrope 6,6,12-graphyne has been predicted to have two nonequivalent types of Dirac points—in contrast to graphene, in which all Dirac points are equivalent—and may therefore have more versatile applications. ... 

Figure 2c shows the electronic structure of graphene described by a simple tight-binding Hamiltonian the electronic wavefunctions from different atoms overlap. However, such an overlap between the Pz(it) orbital and the Px and Py orbitals is zero by symmetry. Thus, the Pz electrons form the 71 band, and they can be treated independently from the other valence electrons. The two sub-lattices lead to the formation of two bands, n and Jt, which intersect at the corners of the Brillouin zone. This yields the conical energy spectrum (Dirac cone, inset in Fig. 2c) near the points K and K, which are called Dirac points. The bottom cone (equivalent to the HOMO molecular orbital) is fully occupied, while the top cone (equivalent to the LUMO molecular orbital) is empty. The Fermi level Ep is chosen as the zero-energy reference and lies at the Dirac point. Consequently, graphene is a special semimetal or zero-band-gap semicondutor, whose intrinsic Fermi surface is reduced to the six points at the corners of the two-dimensional Brillouin zone. 

Graphene lattice and its Brillouin zone. Left lattice structure of graphene, made out of two interpenetrating triangular lattices (ai and aj are the lattice unit vectors). Right corresponding Brillouin zone. The Dirac cones are located at the K and K points... 

Fig. 11 (a) Enthalpy evolution for an 8 atom 2D boron system during an evolutionary structure search. The insets shows the structure of a-sheet (b) the top view and side view of Pmmn-boron (c) the band structure of Pmmn-boron (d) the Dirac cone of Pmmn-boron. 

In a strict sense, the time-evolution generated by the operator (2) is acausal A wavepacket that is initially strictly localized in a finite region of space instantaneously spreads over the whole space. Even for the Dirac equation there are some problems with causality and localization (see, e.g., [5]), but since the propagator of the Dirac equation (the time-evolution kernel) has support in the light-cone, distortions of wave functions and wave fronts can at most propagate with the velocity of light. 

Graphene has a unique and curious band structure which can be approximated by a double cone close to the six Fermi points at the corners of the Brillouin zone (see O Fig. 27-18). Commonly referred to as Dirac electrons, the conduction electrons follow a linear energy-momentum dispersion and have a rather large velocity. 

The band structure of a representative three-dimensional solid (/eft) is parabolic, with a band gap between the lower-energy valence band and the higher-energy conduction band. The energy bands of 2D graphene right) are smooth-sided cones, which meet at the Dirac point... 

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