Valence band width

Here, W is a cut-off of the order of the 7t-band width, introduced because the right-hand side of Eq. (3.13) is formally divergent. As in the discrete model, the spectrum of eigenstates of Hct for A(a)= Au has a gap between -Ao and +Alh separating the empty conduction band from the completely filled valence band. 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

Fig. 3.1. The location of the donor and acceptor levels relatively to the crystal energy bands. The levels of valence band are shaded to underline their occupation, the levels of conductivity band are empty. The conductivity band width Ew and the value 5E (see formula (23) in the text) are shown.

The ionization potentials (IP), electron affinities (EA), fundamental gaps (AEg)t and valence- and conduction-band widths (8EV and 8EC, respectively) of... 

Table 2 Some one-electron properties of the investigated polyacetylene models IP (ionization potential), EA (electron affinity), fundamental gap (AE0)t valence band width (8EV), and conduction band width (8Ec), respectively, obtained from calculations up to the 28th neighbouring C2H2 unit (all values in eV)...

Later the MINDO/2 CO calculation was repeated for the two-dimensional parallel-chain pleated sheet / -polyglycine network (two glycine molecules in the unit cell) using second neighbours interactions.101 These resulted in values of 6ev 1.2 eV, 6ec 1.7 eV and a forbidden band width of 4.8 eV. It should be mentioned that while in the case of the polyformamide we had planar structures and could therefore define n- and -electron bands (in both cases the valence band was a n- and the conduction band a parallel-chain pleated sheet conformation of / -polyglycine because it is not planar. 

The effect, shown in the third panel of Fig. 6-3, accounts for the curvature of the upper set of valence bands (and adds curvature to the others) this the Fponly bands lacked. It does this without modifying the total band width nor the band... 

Poole, Liesegang, Leckey, and Jenkin (1975) have reviewed published band calculations for the alkali halides and tabulated the corresponding parameters obtained by various methods. Pantclidcs (1975c) has used an empirical LCAO method that is similar to that described for cesium chloride in Chapter 2 (see Fig. 2-2), to obtain a universal one-parameter form for the upper valence bands in the rocksalt structure. This study did not assume only one important interatomic matrix clement, as we did in Chapter 2, but assumed that all interatomic matrix elements scale as d with universal parameters. Thus it follows that all systems would have bands of exactly the same form but of varying scale. That form is shown in Fig. 14-2. Rocksalt and zincblende have the same Brillouin Zone and symmetry lines, which were shown in Fig. 3.6. The total band width was given by... 

These values could be subtracted to obtain the difference between valence-band minima on the two sides, but it is the difference between valence-band maxima which is of central interest and which we treated in Section 10-F. We therefore estimate a valence-band maximum as plus the band width (taken from... 

We calculated the position of the valence-band maximum, K,.., ,-f band width, with respect to an artificial metallic Fermi energy in Section 18-G. This may be u.sed to give an estimate of the position of the Fermi energy of a metal contact with respect to the valence-band maximum. Give this estiirmtc for all of the semiconductors listed in Table 18-3, and compare with the empirical estimate of one third the minimum band gap (Table 10-1). One feature of the result is experimentally correct the position is quite insensitive to the work function of the metal. In addition, the trends arc mostly correct, though the values are not accurate. 

The compilation of such data constituted a firm basis that was used to study a specific and more complicated system the elucidation of the electronic structure of a copolymer of ethylene (48%) and tetrafluoroethylene (52%) whose synthesis was conducted in order to maximize the alternating sequences. The valence band spectrum of such a compound (Figure 8) was found very similar to the one measured e.g. for poly(vinylidene fluoride). But, by looking to the fine details of the spectrum, by simulating the valence band of a block copolymer (by addition of PE and PTFE spectra), and by comparison with model calculations, it was possible to show that the C-C band width and the distance F2s-top of the C-C band were characteristic of an ethylene-tetrafluoro-ethylene copolymer with dominant alternant structure (28). 

Isotactic polypropylene does not exist in the two zig-zag and helix conformations, but syndiotactic PP is a good candidate for this search. Indeed, according to the preparation procedure of the polymer ( ), it can exist in a helix or zig-zag planar conformation. Syndio-PP films were prepared following both ways, their conformation checked by IR, and studied by XPS. Their valence band spectra again show distinct differences in the C2s band (Figure 16) for zig-zag PP that is probably highly amorphous, the C-C band width increased by about leV, whereas the bonding subband increased in intensity and became more structured ( ). 

---