Brillouin Zone defined

A more complete view of the electronic structure can be obtained from quantum mechanics. All of this information is contained within the Brillouin zone, but this gives us a conceptual problem it is a complex three-dimensional shape that resides in reciprocal space. We can simplify any three-dimensional shape by cutting slices through it to create two-dimensional representations. In this way, a sphere becomes a circle and so on. We can do the same with the Brillouin zone. By slicing through certain pathways, called k vectors, which link k points (which are special positions in the three-dimensional Brillouin zone defined by the real-space crystal system) we generate the two-dimensional band structure diagrams. 

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. 

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. 

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. 

Fig. 9. Direct band gap for [9,2] nanotube in vicinity of band gap. Wave number is dimensionless coordinate x, with onedimensional Brillouin zone for x defined —tt < x < tt.

Attempts to relate the stability of this phase to the filling of a Brillouin zone were not successful no well-defined zone could be found. 

The reciprocal lattice is useful in defining some of the electronic properties of solids. That is, when we have a semi-conductor (or even a conductor like a metal), we find that the electrons are confined in a band, defined by the reciprocal lattice. This has important effects upon the conductivity of any solid and is known as the "band theory" of solids. It turns out that the reciprocal lattice is also the site of the Brillouin zones, i.e.- the "allowed" electron energy bands in the solid. How this originates is explciined as follows. 

By changing the scattering parameters we can tune in to electrons with a well-defined, constant momentum value along the c-axis. In the present case pc = 0.25 au i.e. the resulting measurement was on the boundary of the Brillouin zone, along the A-L direction. Now two structures are visible, both the ou and the k band. The dispersion of the ou band has not changed noticeably from that found for the T-M... 

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). 

The key features of this integral are that it is defined in reciprocal space and that it integrates only over the possible values of k in the Brillouin zone. Rather than examining in detail where integrals such as these come from, let us consider instead how we can evaluate them numerically. 

The phonons are not stationary modes, but traveling waves extending through the whole crystal. The momentum of a phonon can be assigned as equal to /iq, in analogy with the momentum of a photon, though it is not strictly defined, as the phonon can be described equivalently in an extended Brillouin zone (see Fig. 2.1), corresponding to a different value of the wavevector q. 

It is clear from Eq. (2.2b) that the frequency to in Eq. (2.7) is a function of q, because q governs the relative displacement of two interacting atoms. The co(q) dependence on q (the dispersion relationships) is illustrated in Fig. 2.1 for the rock-salt structure. It can be shown that all normal modes can be represented in the first Brillouin zone, which extends from 0 to nja in the a direction of the rock-salt structure, or, more generally, is bounded by faces located halfway between the reciprocal lattice points in the space defined by1 = 27r<5fJ-. The... 

The frequency vD at the edge of the Brillouin zone is thus equal to vsqD/2n. The Debye temperature 0D is defined as hvD/(kB). As shown below, 0D is an inverse measure for the vibrational mean-square amplitudes of the atoms in a crystal at a given temperature. 

Fig. 6. Energy of an electron in a crystal plotted against wave number kx for a simple-cubic lattice with Brillouin-zone boundaries at kx = (7r/a), zt(2ir/a). Solid line is for reduced wave number — (w/a) < kx < (w/a). Heavy dashed line is for kx defined in range — o < kx < . Light dashed line is solution for a constant potential. Quantum number m defines the band. Effective mass of interest is...

Consider a Brillouin Zone, such as that defined for CsCl, and an energy band (k), defined within that zone. F urther, imagine a single electron within that band. If its wave function is an energy eigenstate, the time-dependent Schroedingcr equation, Eq. (1-17), tells us that... 

We should now relate what we have found in this chapter to the energy-band description introduced initially in Section 2-A the crystal used there also had the translational symmetry of the simple cube. There we also defined wave numbers for the states but restricted their domain to a Brillouin Zone similarly, in Section... 

In the theoretical description of regular polymers, the monoelectronic levels (orbital energies in the molecular description) are represented as a multivalued function of a reciprocal wave number defined in the inverse space dimension. The set of all those branches (energy bands) plotted versus the reciprocal wave number (k-point) in a well defined region of the reciprocal space (first Brillouin zone) is the band structure of the polymers. In the usual terminology, we note the analogy between the occupied levels and the valence bands, the unoccupied levels and the conduction band. 

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