Jones Zone

Polydiacetylene crystals. The enhancement of x because of one-dimensional electron delocalization is strikingly corroborated in the polydiacetylene crystals. Their structure is that of a super alternated chain with four atoms per unit cell and the Huckel approximation yields four bands for the ir-electrons, two valence and two conduction bands. When depicted in the extended Jones zone, each pair can be viewed as arising by a discontinuity at the middle of the Brillouin zone of the polyene chain. The dominant contribution to X(2n 1) comes from the critical point at the edge of the extended Jones zone (initially at the center of the reduced B.Z.). The complete expressions are derived in (4,22) and calculated for different polydiacetylenes. We reproduce the values of x 2 for TCDU and PTS in table IV. The calculated values are in good agreement... 

The largest peak in Fig. 4-3, labelled 2. can be easily identified with the fairly parallel bands separated by between 4 and 6 eV over most of the region shown in Fig. 4-4. (A careful study of this was made recently by Kondo and Moritani, 1977.) These bands arise from the Jones Zone, which will be discussed in detail in the treatment of tetrahedral semiconductors with pseudopotentials in Chapter 18. The energy at which this peak occurs was used earlier as a basis for obtaining experimental values for the covalent energy Fj (Harrison and Ciraci, 1974). 

Part (a) shows a frce-clectron Fermi sphere for silicon cut by various Bragg reflection planes, reducing the area of free Fermi surface. Part (b) shows the Jones Zone, made up of (220) Bragg planes, into which all of the silicon Fermi surface has disappeared. The view is along a [llO] direction in both parts. 

We know that when the pseudopotential is at full strength, all of the Fermi surface must disappear, since none is present in the semiconductor. We can, in fact, see from the figure that what must happen is that it disappears into the slanted and vertical planes of Fig. 18-3,a the horizontal planes as well as the omitted (111) planes are noncssential. Indeed, the vertical and slanted planes are among the twelve (220) Bragg planes that make up the Jones Zone (Mott and... 

Jones, 1936 1958, p. 159), which has long been thought to dominate the electronic structure of the covalent solids. A sketch of that zone is shown in Fig. 18-3,b. The volume contained within it is just sufficient to contain the four electrons per atom, and it would be natural to assume that all Bragg planes other than those bounding the Jones Zone are unimportant and could be neglected in a semiquantitative theory. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

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 the covalent solids, the Jones Zone gap should be identified with the principal optical absorption peak previously identified with LCAO interatomic matrix elements. Thus it allows a direct relation between the parameters associated with the LCAO and with the pseudopotential theories. It is best, however, to simplify the pseudopotential analysis still further before making that identification. 

We found earlier in this ciiapter that this is the dominant matrix element that opens up the gap at the Jones Zone, eliminates tlie Fermi surface, and dominates the bonding properties. As we slowly displace the second sublattice, as in... 

Whereas fee contains 4 and bcc 2 atoms per unit cell, y-brass is rather complex with 52 atoms per cell. Z became close to 1.8 e/a, and the Jones zone is... 

Fig. 5.6. Phase regions of some Hume-Rothery alloys vs. Z and the corresponding Brillouin or Jones zones...

Ion positions in disordered systems are not as well defined as in crystalline systems, and so the Jones zone can adapt to the electronic constraints over large concentration ranges. In this respect, all structure data of amorphous and liquid... 

These bands arise from the Jones Zone, which will be discussed in detail in the treatment of tetrahedral semiconductors with pseudopotentials in Chapter 18. The energy at which this peak occurs was used earlier as a basis for obtaining experimental values for the covalent energy V2 (Harrison and Ciraci, 1974). 

FIGURE 18-4 The ratio (w /Ep) of pseudopotential form factor to free-electron Fermi energy for silicon, showing that the direct Jones-Zone diffraction [220] should be weak. 

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