Energy Gap at the Brillouin Zone

The electron density represented by is concentrated mainly near the ion cores, 

---

We see that the structural trend from fee - bcc hep is driven by the van Hove singularities in the densities of states. These arise whenever the band structure has zero slope as occurs at the bottom or top of the energy gaps at the Brillouin zone boundaries. The van Hove singularities at the bottom of the band gap at X and at the top of the band gap at L in fee copper are marked X4. and Ly, respectively, in the middle panel of Fig. 6.16. It is, thus, not totally surprising that the reciprocal-space representation... 

Now that we have shown that there exists an energy gap at the Brillouin zones, we are in a position to understand the succession of phases from face-centered cubic (fee) to body-centered cubic (bcc) to various types of hexagonal close-packed (hep) phases seen in the Cu-Zn (Figure 19.4) and other systems with mixed valences as discussed in Section 12.6.8. 

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 ... 

The bandstructure of fee aluminium is shown in Fig. 5.9 along the directions and TL respectively. It was computed by solving the Schrodinger equation selfconsistently within the local density approximation (LDA). We see that aluminium is indeed a NFE metal in that only small energy gaps have opened up at the Brillouin zone boundary. We may, therefore, look for an approximate solution to the Schrodinger equation that comprises the linear combination of only a few plane waves, the so-called NFE approximation. 

Alternation of the CC bond lengths along the chain and the existence of a large energy gap are well-established facts in PA (see Chapter 12, Section II.C.2). However, since each carbon atom contributes one tt electron, there is at first sight no obvious reason why CC bonds should not be equivalent. If they were, and taking into account the electron spin, the tt electrons should generate a half-filled band such a material is a metal. If there is bond alternation, the one-dimensional unit cell is doubled and a gap opens at the Brillouin zone boundary the material is a semiconductor. 

The origin of these effects has been debated. One possibility is the Peierls instability [57], which is discussed elsewhere in this book In a one-dimensional system with a half-filled band and electron-photon coupling, the total energy is decreased by relaxing the atomic positions so that the unit cell is doubled and a gap opens in the conduction band at the Brillouin zone boundary. However, this is again within an independent electron approximation, and electron correlations should not be neglected. They certainly are important in polyenes, and the fact that the lowest-lying excited state in polyenes is a totally symmetric (Ag) state instead of an antisymmetric (Bu) state, as expected from independent electron models, is a consequence... 

Thus, the electrons at the Brillouin zone boundary can have two different energies for the same wave vector and thus the same states. One energy value is somewhat lower than the free-electron gas value, the other one is somewhat higher. Energies between these values are unattainable for any electrons - there is now an energy gap in the E = E(k) relation for all fesF vectors ending on the Brillouin zone. 

Cubic ZnS is a direct-gap semiconductor with the smallest energy gap at the center of the Brillouin zone (F). When spin-orbit splitting is taken into account, the topmost valence band state Fi5v splits into Fgv and F7v further splitting into A, B, and C levels is caused by the crystal field. 

The Fermi sphere could expand across the zone boimdary, but remember there is an energy gap at the boundary and this also would cost energy. Is there another configuration that the system could go into that would allow more electrons per atom before the Fermi sphere encounters the first Brillouin zone ... 

The dispersion relation (Eq. (5.34)) yields two bands, which for an infinite ( bulk ) crystal are displayed by the thick lines in the left part of Eigure 5.12. As can be seen, an energy gap of width 2Vg opens up at the Brillouin zone boundary k = f Note that for an infinite crystal the wave vector k has to be real, since otherwise the Bloch wave function F(z) = e" Ut(z) would diverge exponentially for either z -r- +00 or z —00. 

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... 

---