Dirac points

Due to its zero-gap electronic stmcture, large graphene sheets are not suitable for FET applications [185, 193]. It has been shown that graphene transistors conduct current even at the point of expected isolator behavior (i.e., Dirac point) [288]. Therefore, current modulation cannot be achieved using macroscopic graphene sheets [185]. In order to have a band gap on graphene, the use of narrow ribbons of... 

Abanin DA, Novoselov KS, Zeitler U et al (2007) Dissipative quantum hall effect in graphene near the Dirac point. Phys Rev Lett 98 196806... 

Dirac pointed out that the coupling energy proportional to 2Ky si Sj) is much greater than that of the spin magnetic moments. Such models [...] were in use before the justification by... 

Mafra DL, Samsonidze G, Malard LM, Elias DC, Brant JC, Plentz E, Alves ES, Pimenta MA (2007) Deterination of LA and TO phonon dispersion relations of grapehene near the Dirac point by double resonance Raman scattering. Phys Rev B 76 233407... 

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. ... 

Now we are in position to recognize the Poisson summation (the comb) formula (see Appendix A. 1.2) linking the exponential fluctuations with the delta Dirac point-contributions... 

Fig. 7. Left panel Cooper pairs (—k, k) and electron-hole (Peierls) pairs (—k, —k + Q) for the n.n. tight binding Fermi surface (thick line) with perfect nesting vector Q. Saddle points (S) of 6(k) at (0, .7t) and ( 7T, 0) lead to DOS peak at the Fermi energy. Therefore unconventional pair states can only have nodes away from S, i.e., at the Dirac points D ( j, j) where the quasiparticle spectrum takes the form of eq. (54). This is the case for a

Fig. 14. Left Illustration of staggered charge or spin current pattern in the square lattice for d-CDW or d-SDW. Right Magnetoresistance (j T ac-plane) oscillations in the low temperature d-CDW state of the organic quasi-ID conductor Q -(BEDT-TTF)2KHg(SCN)4. The oscillations result from the Landau quantisation of energy levels eq. (55) around the nodal Dirac points in fig. 7. The circles are from experiment (T = IAK, B = 157 , = 45°), and the full line...

Giant diamagnetism The susceptibility has been analysed in detail (Nersesyan and Vachnadze, 1989 Nersesyan et al, 1991). Strong anomalies in the diamagnetic susceptibility for both d-CDW and -SDW are predicted at low fields. This is due to the peculiar conical or relativistic quasiparticle spectrum around the nodal Dirac points (D) in fig. 7. For T the spectrum can be linearized and consists of two bands... 

Finite frequency probes Finally we discuss finite freqnency probes for d-DW states like optical conductivity (Yang and Nayak, 2002). It exhibits non-Drade like behaviomat low frequencies because of arbitrary low excitation energies for q = 0 interband E- Efr) transitiorrs at the nodal (Dirac) points. For perfect nesting Ep = Q) at low temperatures one obtairrs (for (WT 1, r = quasiparticle lifetime)... 

Figure 10.6 (a,b) Electronic band structure of graphene showing valence and conduction bands touching at the Dirac points and having a linear dispersion relation around the Fermi level. Comparison of (c) zero-band gap graphene, (e) band gap-opened... 

G., Kim, B.S., Kang, D.J., and Shin, H.S. (2012) Reversibly light-modulated dirac point of graphene functionalized with spiropyran. ACS Nano, 6, 9207 - 9213. 

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. 

Using this method, dual-gated field-effect transistors were fabricated on Si02/Si substrates and an electron mobility as high as 4050 cm /V s with the residual carrier concentration at Dirac point of Hq = 3.2 X 10 cm-2 at room temperature was achieved. 

Figure 6.88. Band structure and the ambipolar field effect in graphene. Conduction and valence bands meet at the Dirac point without an external field in the presence of a gate bias, the Fermi level moves above/below the Dirac point to generate free carriers. Reproduced with permission from Nature Mater. 2007, 6, 183. Copyright 2007 Nature Publishing Group.

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... 

The two points K and K at the corners of the graphene Brillouin zone (BZ) are named Dirac points. Their positions in momentum space are given by... 

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