Band structure diamond crystal

Silicon crystallizes in the diamond structure,16 which consists of two interpenetrating face-centered cubic lattices displaced from each other by one quarter of the body diagonal. In zinc blende semiconductors such as GaAs, the Ga and As atoms lie on separate sublattices, and thus the inversion symmetry of Si is lost in III-V binary compounds. This difference in their crystal structures underlies the disparate electronic properties of Si and GaAs. The energy band structure in... 

Silicon is a semiconductor with an intrinsic conductivity of 4.3 x 10" Q" cm and a band gap of I.I2eV at 300K. It has a diamond crystal structure characteristic of the elements with four covalently bonded atoms. As shown in Fig. 2.1, the lattice constant, a, is 5.43 A for the diamond lattice of silicon crystal structure. The distance between the nearest two neighbors is V3a/4, that is, 2.35 A, and the radius of the silicon atom is 1.18 A if a hard sphere model is used. Some physical parameters of silicon are listed in Table 2.1. 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

Table 3.4 gives a few relevant band-structure parameters of group-IV and group-III V crystals. The structure of the CB near from its minimum is generally simpler to model than that of the VB. The CB parameters are known, therefore, with a reasonable accuracy from the experimental data. For diamond, mnt = 0.31 me is deduced from the Zeeman splitting of 2p (P) in Cdiam [19] and mn = 1.7 me from the ratio 7 = mnt/mn whose determination is explained in the text accompanying Table5.3. 

A Figure 12.30 The electronic band structure of semiconductors that have the diamond crystal structure. 

Silicon and germanium are of great importance as intrinsic semicondnctors. Being unable to form double bonds, they can only crystallize in the cubic diamond structure (Figure 4a) with jp -hybridized atoms linked by four identical covalent bonds. Of particular interest is their band structure (Figure 5a). In diamond the C-C bond is so strong (C-C = 1.54 A) that, in spite of the band spreading, the... 

Using the same tight-binding model as in the previous problem, calculate the band structure of polyethylene and of the diamond crystal and compare the two. Obtain the density of states for polyethylene with sufficient accuracy to resolve the van Hove singularities as discussed in chapter 5. 

Pig. 2.17 Band structure of AB crystals of the zinc-blende type. The rvth main group — from top to bottom — is represented in the upper row (diamond structure if A=B). In the right hand column the vertical distance in the periodic table is increased iso-electronically. According to Ref. [31]. 

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