Brillouin zone corner

All other a-phase salts (M = K, Rb, and Tl) showed AMRO periodic in tan 0, too. In contrast to the magnetoresistance shape in the former materials no resistance maxima as expected for a 2D warped cylinder but instead sharp resistance minima were observed [289, 309, 310, 311]. Figure 4.14a shows examples of this behavior for Q -(ET)2KHg(SCN)4 and a-(ET)2RbHg(SCN)4 at R = 11T and T — 1.4K [312]. Very prominent minima periodic in tan are visible in the magnetoresistance. These minima cannot be explained by the 2D warped FS model but are attributed to Lebed-like oscillations of the ID bands [313] (see also Sect. 3.3). From the calculated band structure shown in Fig. 2.20 one can see that the FS of the a phase consists of 2D closed orbits in the Brillouin zone corner and two ID open sheets perpendicular to the a direction. The maximum amplitude of the AMRO, however, was not observed for a rotation of the field parallel to these sheets but for angles 20° and 24° towards the a direction of the M = K and M = Rb crystals, respectively (see Fig. 4.14a). 

These surprising results can be understood on the basis of the electronic structure of a graphene sheet which is found to be a zero gap semiconductor [177] with bonding and antibonding tt bands that are degenerate at the TsT-point (zone corner) of the hexagonal 2D Brillouin zone. The periodic boundary... 

The volume of Brillouin zone Vg corresponds to (IttY times the reciprocal of cell volume 17, commonly adopted in crystallography, and the origin of the coordinates is at the center of the cell (and not at one of the corners) ... 

Figure 39 Band structure of the (a) fully saturated and (b) partially saturated Si[i]-SiC>2(0 01) SL projected along the two symmetry directions of the 2D Brillouin zone of the (0 01) surface. K and M represent, respectively, the k-points in the corner and in the middle of the side of the 2D Brillouin zone. A self-energy correction of 0.8 eV has been added to the conduction states. Energies (in eV) are referred to the valence band maximum.

Cristobalite undergoes a first-order displacive phase transition at around 500 K at ambient pressure (Schmahl et al. 1992, Swainson and Dove 1993, 1995a). The distortion of the structure can be associated with a RUM with wave vector (1,0,0), which is at the corner of the Brillouin zone (Dove et al. 1993, Swainson and Dove 1993, 1995a Hammonds et al. 1996). At ambient temperature there is another first-order displacive phase transition to a monoclinic phase on increasing pressure (Palmer and Finger 1994). A recent solution of the crystal structure of the monoclinic phase (Dove et al. 2000c) has... 

As the Fermi surface begins to approach the boundaries of the first Brillouin zone, however, it will become distorted. In directions such as OP (fig. 5.10 d) the energy is abnormally depressed, for a given value of k, and the Fermi surface is correspondingly distended (contour (3) in fig. 5.13). When the surface finally touches the boundary of the Brillouin zone further growth in that direction is arrested and the Fermi surface then extends only towards the corners of the zone (contours (4-6) in fig. 5.13). If the system is one corresponding to the condition shown in fig. 5.12a the zone will ultimately be fully occupied and... 

It has been predicted theoretically [21] and demonstrated experimentally [22] that one or a few layers of atomically flat graphene exhibits two-dimensional semimetal properties with a small overlap (ca. 0.04 eV) between the valence and conductance bands at six symmetric points in the corner of the Brillouin zone, as shown in Figure 14.4. Thus, a nonzero density of states is found at the Fermi level, although the Fermi surface consists of only isolated points. This is attributed to the delocalization of electrons in the graphene plane and results in high in-plane conductivity. Across the plane, there is little interaction, and thus it shows very small conductivity. 

Fig. 2 (Colour online) a) Crystal lattice of graphene. This is composed by two sub-lattices defined by the atoms A and B b) Brillouin zone of graphene, showing the high-symmetry points and the reciprocal vectors c) Electronic structure of graphene calculated by tight-binding close to K and K points the electronic structure is linear. This gives a cone of carriers" at the corners of the 2D Brillouin zone (inset).

If we now move from real space into reciprocal space, the Brillouin zone associated with the crystal lattice is also hexagonal and it shows characteristic high-symmetry points the centre is called F point, while two consecutive corners are denoted as K and K points. Fig. 2b. 

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 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 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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