Brillouin zones energy bands

Fig. 5.12. (a) Variation of electron energy with wave number for a two-dimensional metal on the basis of the Bloch theory. ka, kb and kc are values of the first forbidden wave number in different directions in the first Brillouin zone. The band of forbidden energy in (b) indicates that the energy levels of the first and second zones do not overlap, (c) As (a) but for the case in which there is overlap between the energy levels of the first and second zones, as shown in ([Pg.99]

At this stage one should mention some important quantities which may be derived immediately from the energy bands. The density of states in reciprocal space is uniform, and in the Brillouin zone each band has one state per... 

Electrons are found in these energy bands called permitted bands or Brillouin zones.. The band in which valence electrons are present is called valence band. The outer empty band is called conduction band. 

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.

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. 

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... 

Figure 4 Conduction band levels and excitation levels of infinite periodic hydrogen chains by using different approximations of the polarization propagator. The left part refers to the crystalline orbital energy differences, namely, the Hartree-Fock excitation energies the right part refers to the random phase approximation results obtained by using 41 k-points in half the first Brillouin zone.

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. 

In white tin, the nearest neighbor interatomic distance is increased to about 3 A. As expected, this lowers the energy of the 5s anti-bonding functions, and in a considerable portion of the Brillouin zone the 5 antibonding functions lie below the 5p like functions in energy hence, there are substantially more than one 5s electron and substantially less than three 5p electrons per atom. Judging from the Mdssbauer data (one could get this presumably from the band calculations with enough effort). 

In addition to the acoustical modes and MSo, we observe in the first half of the Brillouin zone a weak optical mode MS7 at 19-20 me V. This particular mode has also been observed by Stroscio et with electron energy loss spectrocopy. According to Persson et the surface phonon density of states along the FX-direction is a region of depleted density of states, which they call pseudo band gap, inside which the resonance mode MS7 peals of. This behavior is explained in Fig. 16 (a) top view of a (110) surface (b) and (c) schematic plot of Ae structure of the layers in a plane normal to the (110) surface and containing the (110) and (100) directions, respectively. Along the (110) direction each bulk atom has six nearest neighbors in a lattice plane, while in the (100) direction it has only four. As exemplified in Fig. 17, where inelastic... 

Fig. 29. Dispersion curves of some features of the valence band of UO2 (see Table 3) as obtained by angular resolved photoemission. The Brillouin zone plane examined is the rXT X plane. Data are for an excitation energy hv = 40.8 eV (Hell) squares are for an excitation energy hv = 21.2 eV (He I) (from Ref. 15)...

Fig. 1. Representation of the band structure of GaAs, a prototypical direct band gap semiconductor. Electron energy, E> is usually measured in electron volts relative to the valence, v, band maximum which is used as the zero reference. Crystal momentum, k, is in the first Brillouin zone in units of 2%/a...

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