Brillouin zones

Because (k) = (k + G), a knowledge of (k) within a given volume called the Brillouin zone is sufficient to detennine (k) for all k. In one dimension, G = Imld where d is the lattice spacing between atoms. In this case, E k) is known once k is detennined for -%ld < k < %ld. (For example, m the Kronig-Peimey model (fignre Al.3.6). d = a + b and/rwas defined only to within a vector 2nl a + b).) In tlnee dimensions, this subspace can result in complex polyhedrons for the 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.

Fig. 3.16 The efect of introducing a weak potential into the ID lattice is to lift the degeneracy of the energy levels mar to the edge of the Brillouin zone (shown in both extended-zone and reduced-zone representation).

Fig. 3.17 The two possible sets of standing waves at the Brillouin zone boundary. Standing wave A concentrates electron density at the nuclei, whereas wave B concentrates electron density between the nuclei. Wave A thus has a lower energy than wave B.

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

I be second important practical consideration when calculating the band structure of a malericil is that, in principle, the calculation needs to be performed for all k vectors in the Brillouin zone. This would seem to suggest that for a macroscopic solid an infinite number of ectors k would be needed to generate the band structure. However, in practice a discrete saaipling over the BriUouin zone is used. This is possible because the wavefunctions at points... 

Monkhorst H J and J D Pack 1976. Special Points for Brillouin-zone Integration. Physical Review B13 5188-5192. 

Once the phonon frequencies are known it becomes possible to determine various thermodynamic quantities using statistical mechanics (see Appendix 6.1). Here again some slight modifications are required to the standard formulae. These modifications are usually a consequence of the need to sum over the points sampled in the Brillouin zone. For example, the zero-point energy is ... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

The electronic structure of an infinite crystal is defined by a band structure plot, which gives the energies of electron orbitals for each point in /c-space, called the Brillouin zone. This corresponds to the result of an angle-resolved photo electron spectroscopy experiment. 

Simply doing electronic structure computations at the M, K, X, and T points in the Brillouin zone is not necessarily sufficient to yield a band gap. This is because the minimum and maximum energies reached by any given energy band sometimes fall between these points. Such limited calculations are sometimes done when the computational method is very CPU-intensive. For example, this type of spot check might be done at a high level of theory to determine whether complete calculations are necessary at that level. 

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

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... 

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... 

Figure 3.27. The first Brillouin zone of the facc-ccntrcd cubic structure, after Pippard.

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 Born-von Karman boundary conditions then restrict the allowed electronic states to those in the graphene Brillouin zone that satisfy... 

Fig. 3. (a) Depiction of central Brillouin zone and allowed graphene sheet states for a [4,3] nanolube conformation. Note Fermi level for graphene occurs at K points at vertices of hexagonal Brillouin zone, (b) Extended Brillouin zone pie-ture of [4,3] nanotube. Note that top left hexagon is equivalent to bottom right hexagon. 

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. 

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