Reduced-zone scheme

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

Equation (4) holds generally at the face center but is valid over the whole face if the crystal point group contains a reflection plane through the zone center that is parallel to the face. It also holds for all k vectors that terminate on a line in the BZ face that is parallel to a binary axis. The E(k) may be described either by a singlevalued function of k (with k > 0), which is called the extended zone scheme, or by a multivalued function of k within the first BZ, the reduced zone scheme (see Figure 17.2). 

Figure 17.2. E(k) for the 1 -D free electron model in (a) the extended zone scheme and (b) the reduced zone scheme.

Find expressions for, and sketch in the reduced zone scheme, FE energy bands along TH in the reciprocal lattice of the bcc lattice. 

PROBLEM 8.6.1. A fictional simple cubic crystal has a lattice constant a = 4.21 A. Compute the four lowest free-electron energy levels along the wavevector k in the reduced zone scheme at the fc-space point (n/2a, 0, 0). 

The requirement that — 1 < cos ka< 1 used to determine the allowed bands means that the value for k for each band is not unique in the way implied by the extended band scheme shown in Fig. 4.4(b). Instead, the band structure is more properly represented in a reduced zone scheme with —n/a < k < nja, which corresponds to the first Brillouin zone, as shown in Fig. 4.4(c). 

Tlie displacement of regions of wave number space so tliat each state is represented by a wave number inside the Brillouin Zone equivalent to its wave number outside. The reduced-zone scheme in part (b) gives the second band Fermi surface in the ordinary band dcseription in the Brillouin Zone. 

If we wish to study a stale at the face of the Jones Zone, we must consider not only the plane wave with wave number at that face, say k,, o = [110]27c/a, and that at the opposite face, /states differs from the others by a lattice wave number, so that if the free-clectron bands were plotted in the reduced-zone scheme, they would all be at the same point, the point [001]27c/fl, which is at the center of one of the square faces of the Brillouin Zone, for example, the point X in... 

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. 

Ea k) is shown in Fig. 7 in a reduced zone scheme (231). There are a total of N/2 states in each of two bands of width = 2 ] Ia21 and an energy gap between them of Eg = 2( /<,i( — l osj). Each state may accommodate two electrons (of opposite spin), allowing iV electrons per band or IN electrons for both bands together. Examining Eq. 15, when a equals 02 (or equivalently, when tai equals tc, the energy bands for the dimer case become identical with the results derived for the uniform chain, Eq. 11. 

Fig. 7. Energy bands for a dimerized chain in a reduced zone scheme (231) ...

We can find the dispersion curves for the two modes from Eq. (15). Either extended or reduced zone schemes can be used. Analogous to the band gap in the electronic band structure, the dispersion curve for the negative mode has a gap between X ... 

The classical theory of the dielectric response in solids is frequently described by the Drude and Lorentz models. The Drude model is applicable to free-eiectron metals its quantum-meehanical analog includes intraband transitions, where intraband transitions are taken to mean all transitions not involving a reciprocal lattice vector. The Lorentz model is applicable to insulators its quantum-mechanieal analog ineludes all direct interband transitions, i.e., all transitions for whieh the final state of an electron lies in a different band but with no change in the k vector in the reduced-zone scheme. In the following discussion, both models will be surveyed and evaluated for real metals. 

The optical and acoustic modes for NaCl were computed from the elastic coefficients in Table 16.1 and displayed in the reduced zone scheme in Figure 16.6. As k increases beyond Ti/2a, the dispersion curve is the same as for —kirfa therefore, all the pertinent information is contained in the reduced zone from 0 > kir/Ta. These optical and acoustic modes can be mapped by neutron inelastic scattering. 

Dispersion relaKons for NaCl in the reduced zone scheme for the various modes of oscillation longitudinal optical (LO), transverse optical (TO), longitudinal acoustic, and transverse acoustic. The curves were computed from Equation 16.19 using the elastic coefficients for NaCl given in Table 16.1. It should be noted that the a in this figure is the distance between the individual ions. Usually this dispersion relation is shown plotted between —-ir/fl and -ir/fl where a is taken as the distance between ion pairs. 

The two 3-functions in Eq. (3.2.2.19), providing a Idnematical description of the photoemission process, represent a very stringent condition under which photoemission intensity can be observed, especially when many-body interactions are neglected, and + E - Ei = eB(kj) represents the single-particle band structure of the solid according to Koopman s theorem. The convenient picture of direct transitions arises [19] in a band structure plotted in the reduced zone scheme, the conservation of wave vectors up to a reciprocal lattice vector means that an electron is excited vertically, that is, at constant crystal momentum, from an initial-state band to a final-state band eB(k/) —> C (kj) with energy levels separated by the photon energy hv. In the free-electron final-state approximation, the upper level is... 

The principal properties of the energy as a function of k are as follows. Within the Brillouin zone E(k) is a continuous function. It is, of course, a multiple-valued function in the reduced zone scheme. At a Brillouin zone plane the gradient of (k) must be in the plane, except in certain exceptional cases. Finally, the band structure must be symmetric under inversion, k — k. This is usually referred to as time-reversal symmetry, but for simple Hamiltonians without spin-orbit coupling it follows simply from complex conjugation of Schrodinger s equation. 

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