Symmetry Brillouin zone

In fignre A1.3.9 the Brillouin zone for a FCC and a BCC crystal are illustrated. It is a connnon practice to label high-synnnetry point and directions by letters or symbols. For example, the k = 0 point is called the F point. For cubic crystals, there exist 48 symmetry operations and this synnnetry is maintained in the energy bands e.g., E k, k, k is mvariant under sign pennutations of (x,y, z). As such, one need only have knowledge of (k) in Tof the zone to detennine the energy band tlnoughout the zone. The part of the zone which caimot be reduced by synnnetry is called the irreducible Brillouin zone. 

Figure C2.16.5. Calculated plots of energy bands as a function of wavevector k, known as band diagrams, for Si and GaAs. Indirect (Si) and direct (GaAs) gaps are indicated. High-symmetry points of the Brillouin zone are indicated on the wavevector axis.

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

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. 

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. 

The two peaks at E and E2 are related to transitions (indicated by arrows in Figure 4.7) from the vicinity of specific points of high symmetry in the Brillouin zone, denoted by L and X. However, in order to explain the exact locations of these two absorption peaks, the density of states function must be also taken into account. Thus, it appears obvious that the full interpretation of interband spectra is far from simple. 

Although the number of atoms in a crystal is extremely high, we can imagine the crystal as generated by a spatial reproduction of the asymmetric unit by means of symmetry operations. The calculation can thus be restricted to a particular portion of space, defined as the Brillouin zone (Brillouin, 1953). 

Understanding how symmetry is used to reduce the number of k points for which calculations are actually performed can help in understanding how long individual calculations will take. But overall convergence is determined by the density of k points in the full Brillouin zone, not just the number of k points in the irreducible Brillouin zone. 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

The Brillouin zones for the fee and bcc lattices are drawn in Fig. 5.8. We see that it is customary to assign high symmetry points with specific symbols. Thus, the centre of the Brillouin zone is labelled , whereas the intersections of the 100> and 111> directions with the zone boundary are labelled X and L in the fee case, H and P in the bcc case respectively. 

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