Solids Brillouin zones

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... 

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... 

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. 

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. 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

Figure 4. Momentum density anisotropy of Cu the solid line marks the boundary of the first Brillouin zone solid and dashed contour lines mark positive and negative anisotropies, respectively.

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

Brillouin zone integrations, pervasive in any solid-state calculation, are best performed with the Monkhorst-Pack scheme [6]. These integrations are essentially equally demanding in any representation and with any basis set. This satisfies criterion 5. 

Application of Eq. (2.58) to calculate the temperature factors requires knowledge of the full frequency spectrum of the crystal throughout the Brillouin zone. Such information is only available for relatively simple crystal structures such as Al, Ni, KC1, and NaCl (Willis and Pryor 1975, p. 13ff.). Agreement between theory and experiment for such solids is often quite reasonable. 

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

Chromatic aberrations do not arise in the acoustic microscope because in its usual mode of operation it may be considered essentially monochromatic. Even when it is necessary to take the spread of frequencies in the acoustic pulses into account, the media through which the waves pass are essentially non-dispersive in solids over the frequency range of interest the phonons are very near the centre of the first Brillouin zone where the dispersion relationship is linear, especially for sapphire. 

The relation of the Brillouin zone to the crystal structure is discussed in many textbooks on solid-state physics (see e.g. Jones 1960, Ziman 1964, p. 12, Ashcroft and Mermin 1976, Chap. 8). 

Figure 1. The dispersion of spin excitations near Q along the edge of the Brillouin zone. The dispersion was calculated in a 20x20 lattice for x = 0.06 and T = 17 K (filled squares, the solid line is the fit with u>k = [wq + c2(k — Q)2]1/2). Open squares are the experimental dispersion [1] of the maximum in the frequency dependence of the odd x"(qw), q = k — Q, in YBa2Cua06.5 (x 0.075 [13]) at T = 5 K.

Figure 4 The Brillouin zone of the square lattice. Solid curves are two of four ellipses forming the Fermi surface at small x. Dashed lines are the Fermi surface contours shifted by (k — Q). Regions of k and k + k contributing to the damping (14) are shaded.

Fig. 4 a, b Energy bands of a orientationally ordered simple-cubic solid C60 and b of the fee solid C60 shown in the same simple-cubic Brillouin zone. The generalized tight-binding method is used [20]... 

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...

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