Hexagonal lattice Brillouin zone

It is known that a metallic ID system is unstable against lattice distortion and turns into an insulator. In CNTs instabilities associated two kinds of distortions are possible, in-plane and out-of-plane distortions as shown in Fig. 8. The inplane or Kekuld distortion has the form that the hexagon network has alternating short and long bonds (-u and 2u, respectively) like in the classical benzene molecule [8,9,10]. Due to the distortion the first Brillouin zone reduees to one-third of the original one and both K and K points are folded onto the F point in a new Brillouin zone. For an out-of-plane distortion the sites A and B are displaced up and down ( 2) with respect to the cylindrical surface [11]. Because of a finite curvature of a CNT the mirror symmetry about its surface are broken and thus the energy of sites A and B shift in the opposite direction. 

See Scanning tunneling spectroscopy Superconductors 332—334 Surface Brillouin zone 92 hexagonal lattice 133 one-dimensional lattice 123, 128 square lattice 129 Surface chemistry 334—338 hydrogen on silicon 336 oxygen on silicon 334 Surface electronic structures 117 Surface energy 96 Surface potential 93 Surface resonance 91 Surface states 91, 98—107 concept 98... 

I shall introduce here a concept that is common to all methods. The Brillouin zone (BZ) is described in most solid state physics textbooks. It is defined as a reciprocal lattice cell bounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points. Fig. 2 illustrates the first BZ cell for a hexagonal lattice. 

Fig. 3.16 First Brillouin zone for a simple cubic lattice (a), for a hexagonal lattice (b), and their relationships (c)-(f).

Figures 4.13 and 4.14 present the Brillouin zones for cubic crystal lattices. One can see the Brillouin zone for hexagonal close-packed crystal lattice in Figure 4.15.

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

Figure 4.15 First Brillouin zone of the hexagonal close-packed crystal lattice, kx.ky, kz are the axes of the Cartesian coordinate system in fe-space. The symmetry points and symmetry lines are indicated. See Table 4.1 for details.

Fig. 9.4. Construction of the First Brillouin Zone (FBZ) as a Wigner-Seitz unit cell of the inverse lattice in 2D. The circles represent the nodes of the inverse lattice. We cut the lattice in the middle between the origin node W and all the other nodes (here it turns out to be sufficient to take only the neatest and the next nearest neighbours) and remove all the sawn-off parts that do not contain JV. Finally we obtain the FBZ in the form of a hexagon.

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. 

Fig. 3.6. Primitive unit cell and Brillouin zone for hexagonal lattice...

Fig. 6.1. Supercell (a) and small Brillouin zone (b) for hexagonal plane lattice of boron nitride BN. Atoms 1—18 belong to the superoell, the atoms with the primed numbers are supposed to be identical with the corresponding atoms, belonging to the supercell.

Fig. 63. (a) The hexagonal AlBj type crystal structure of the RGaj compounds. Spheres with and without pattern show the Ga atoms and the R atoms, respectively (b) Brillouin zone of the body-centered tetragonal crystal lattice for da > -Jl. 

A similar construction called a Wigner-Seitz cell can be formed in direct space by taking perpendicular bisectors between each lattice point. The polyhedra formed will fill the 3D space and contain the volume associated with that particular lattice point. From this analogy, the Brillouin zones can be considered Wigner-Seitz cells in reciprocal space. The 3D Brillouin zones for the bcc, fee, and hexagonal close-packed (hep) reciprocal lattices are shown in Figure 6.7. 

Brillouin zones for fee direct lattice (bee reciprocal) (a), bcc direct lattice (fee reciprocal) (b), and hexagonal direct lattice (hexagonal reciprocal) (c). The letter F designates the origin and the other letters designate directions of high symmetry. This notation will be seen later in the band structure of various materials. 

Find the reciprocal lattice vectors for a hexagonal lattice and verify the size of the hexagonal Brillouin zone shown in Figure 6.7c. 

Figure 34 (a) Crystal structure of hexagonal WC. (b) First Brillouin zone for the hexagonal translational lattice corresponding to the hexagonal WC structure. (From Ref. 91. Reproduced with the permission of the American Physical Society, American Physical Society.)... 

Fig.3 7. a) Unit cell and primitive translation vectors of the hexagonal lattice. The Figure also shows the structure of 3-AgI (Wurtzite structure) ( ) Ag-ions, ( ) I-ions, (====) Ag-I bonds ions number 1 to 4 belong to the unit cell, b) Brillouin zone of the hexagonal lattice [3.5]... 

Figure 2.5 (a) The method for construeting the Brillouin zone for a square planar lattice and the first Brillouin zone for (b) a faee-eentered eubic crystal and (c) a body-centered cubic crystal. For the fee erystal the diamond-shaped faees of the Brillouin zone are along cube axes, [100]-type direetions, while the hexagonal faees are along [111] cube diagonals. For discussion of the [100], [111], and other erystal indiees, see Chapter 4. 

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