Brillouin zone boundary

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.

In the left panel of Figure 8 we show the band structure calculation of graphite in the repeated zone scheme, together with a drawing of the top half of the first Brillouin zone. The band structure is for the 1 -M direction. As the dispersion is very small along the c-axis we would find a similar result if we add a constant pc component to the line along which we calculate the dispersion [17]. The main difference is that the splitting of the a 1 and % band, caused by the fact that the unit cell comprises two layers, disappears at the Brillouin zone boundary (i.e. if the plot would correspond to the A-L direction). 

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 ... 

Fig. 5.8 The fee and bee Brillouin zones. The symbol labels the centre of the zone. The intersections of the 100> and 111 > directions with the Brillouin zone boundary are labelled X and L in the fee case and H and P in the bee case.

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. 

Fig. 5.10 The upper figure shows the electron density corresponding to the eigenfunctions and at the fee Brillouin zone boundary, X. We see that piles up charge into the core regions where the potential, V(x), in the middle panel is large and negative. V(x) is the averaged potential over all the atoms in the yz plane of the fee lattice shown in the lower figure.

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... 

Therefore, as shown in Fig. 7.1(a), the bottom of the band is at the centre of the Brillouin zone (0,0,0), whereas the top of the band is at the zone boundary ( / )( 1,1,1), since ssa < 0. It follows from eqn (7.1) that the bottom and top of the band correspond to perfect bonding and antibonding states, respectively, between all six neighbouring atoms, so that the width of the s band is 2 6ssoj, as expected. The corresponding density of states is shown in Fig. 7.1(b). The van Hove singularities, arising from the flat bands at the Brillouin zone boundaries, are clearly visible. 

Discuss the origin of the Hume-Rothery electron phases within the framework of Jones original rigid-band analysis. How does second-order perturbation theory help quantify Mott and Jones earlier supposition on the importance of the free electron sphere touching a Brillouin zone boundary ... 

Fig. I, la) Energy plotied against wave number for ihe free electron model, (b) Energy plotted against wave number for the "quest-free" electron model, showing energy discominuilies at Brillouin zone boundaries...

For many metals, the "nearly free" electron description corresponds quite closely 10 the physical situation. The Fermi surface remains nearly spherical in shape. However, it may now he intersected by several Brillouin zone boundaries which break the surface into a number of separate sheets. It becomes useful to describe the Fermi surface in terms not only of zones or sheets filled with electrons, but also of zones or sheets of holes, that is. momentum space volumes which are empty of electrons. A conceptually simple method of constructing these successive sheets, often also referred lo as "first zone. "second zone." and so on was demonstrated by Harrison. An example of such construction is shown in Fig. 2. 

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...

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... 

To select the tie points, draw a vector through P parallel to n. It cuts the dispersion surface at two points, A on the a branch and B on the branch A and B are the required tie points. We see, therefore, that the outside incident wave gives rise to four waves inside the crystal Koc and Ko(3 in the forward direction (transmitted waves) and two strongly diffracted waves Kgo, and Kg. This is precisely what we would expect from the splitting of the energy at the Brillouin zone boundary. 

The orientational ordering has also been found in other TMTSF compounds having asymmetric anions, such as Re04, NO, FSO, [76]. In these materials the one-dimensional axis, a-axis is doubled by the orientational ordering. Therefore the anion ordering causes a metal-insulator transition because the band gap at the Brillouin zone boundary opens just at the Fermi level. 

While a one-dimensional model is not very realistic, the analytical result (4.39) shows an important feature of a general nature—the fact that the phonon spectrum is bound There are no modes of frequency larger than 2 )o. Note that this upper bound is associated with wavevectors at the Brillouin zone boundary, tliat is, wavelengths comparable to the interatomic distance. 

In disordered metals, the electrons couple with the static structure at any. K-value and are heavily damped close to pseudo Brillouin-zone boundaries. A definite dispersion relation does not exist (Fig. 5.3bl) and Bloch s theorem is no longer valid. Electronic states cannot be described as eigenstates of the system and, strictly speaking, cannot be translated to the reduced-zone scheme. 

Electronic transport properties are strongly influenced by a touch of the Fermi sphere with the zone boundary, in the crystalline as well as in the disordered state. Exhaustive reviews on this subject have been given by Massalski and Mizutani [5.35] and Mizutani [5.20], In the same way as sharp zone boundaries in crystalline materials are responsible for umklapp processes, in amorphous systems we can talk in terms of diffuse umklapp processes caused by the pseudo Brillouin-zone boundary. This description was first introduced by Hafner [5.36]. 

Accordingly, we discuss below the MDOS as structure induced as proposed by several authors [5.18,28,30-33]. The large dots in Fig. 5.15 indicate binding energies of electronic states at the pseudo Brillouin-zone boundary, roughly estimated with the E versus k relation of free electrons... 

Fredrickson and Helfand [58] used a selfconsistent Hartree approximation in Eq. (199) to study the Gaussian fluctuations around the solution, Eq, (200). While for ordinary second order transition the local function <4>2(r) > — << >>2 stays small even at Tc, this local fluctuation diverges here as T - Tc The reason is that normally the phase space for critical fluctuations is only the vicinity of a point in reciprocal space (the surroundings of q = 0 for a ferromagnet, the surroundings of a few discrete points qB at the Brillouin zone boundary for antiferromagnets, etc.), while here it is the vicinity of a sphere ( q = q ). Fluctuations lead here to a divergence of the mean square displacement of 4> similar as it happens due to phonons in one-dimensional crystals. 

Another interesting result from this simple picture of lattice dynamics for an alkali-halide-type crystal is that when the atoms have the same mass. Mg = Mh = M, and there are no differentiating forces between the atoms (such as different next-nearest-neighbor interactions), then G and H are identical within the context of this dynamical model. The crystal is then indistinguishable from a monatomic crystal that has a unit cell length a = all and corresponding Brillouin zone boundary at = -kIu = I-kIu. (See Fig. 6.) The consequence for the lattice dynamics is that the optical and acoustic mode frequencies become the same at = +7r/a, to = If lM), and there is no aeoustic-optical band gap just as in Fig. 4 (lower panel). From a... 

Figure 6. Dispersion relation for an isobaric crystal, where Mq = M. When /j = /2 for this force constant model, the diatomic crystal is equivalent to a monatomic crystal with lattice spacing a 2 or Brillouin zone boundary at 2ir/a. Hence the optical branch (solid line) appears as the portion of the acoustic branch from via to 2ir/a (dashed line) folded back to the zone center from the real Brillouin zone boundary (dotted vertical line).

Incoherent neutron inelastic scattering (INIS) provides information on the density of states (see Appendix) of a crystal which, for the most part, is determined by Brillouin-zone boundary energies of the dispersion curves. 

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