First Brillouin zone description

In the theoretical description of regular polymers, the monoelectronic levels (orbital energies in the molecular description) are represented as a multivalued function of a reciprocal wave number defined in the inverse space dimension. The set of all those branches (energy bands) plotted versus the reciprocal wave number (k-point) in a well defined region of the reciprocal space (first Brillouin zone) is the band structure of the polymers. In the usual terminology, we note the analogy between the occupied levels and the valence bands, the unoccupied levels and the conduction band. 

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. 

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]. 

---