Understanding Band-Structure Diagrams

For lattices with more than one atom per lattice point, combinations of Bloch sums have to be considered. In general, the LCAO approach requires that the result be the same number of MOs (COs in sohds) as the number of atomic orbitals (Bloch sums in sohds) with which was started. Thus, expressing the electron-wave functions in acrystaUine sohd as linear combinations of atomic orbitals (Bloch sums) is really the same approach used in the 1930s by Hund, Mulliken, Htickel, and others to construct MOs for discrete molecules (the LCAO-MO theory). 

It has been shown in the previous chapter how a MO-like treatment of the smallest repeating chemical point group, or lattice point, can be used to approximate the relative locations of the bands at the center of the BZ. To reinforce this idea, repeat the analogy with yet another example. Consider the energy level diagram for a hypothetical 

Some important points to remember, when looking at any band-structure diagram, are  

A band-structure diagram is a map of the variation in the energy, or dispersion, of the extended-wave functions (called bands) for specific Ar-points within the first BZ (also called the Wigner-Seitz cell), which is the unit cell of Ar-space. 

The total number of bands shown in a band-structure diagram is equal to the number of atomic orbitals contributed by the chemical point group, which constitutes a lattice point. As the full crystal structure is generated by the repetition of the lattice point in space, it is also referred to as the basis of the stmcture. 

---

Now consider the solid. It is easy to understand why pure bulk C6o is semiconducting. The clusters are about 3.1 A apart and the interaction between clusters must be small. Therefore, the discrete levels in the HOMO-LUMO region of C6o give rise to narrow or flat occupied bands separated from flat unoccupied bands as shown by the band structure diagram in Figure 7.33 where the first Brillouin zone is shown at the right. Solid C6o is a molecular solid. 

Now we understand the structure of the semiconductor/electrolyte interface, we can draw a detailed band diagram for a complete PEC cell. An example is shown in Fig. 2.16 for a cell composed of a n-type photoanode and a metal cotmter electrode. As usual, the y-axis represent the energy of an electron at a certain point x in the cell. The energy of an electron in vacuum at infinite distance is chosen as a reference. It is important to note that the vacuum level bends in the presence of an... 

The electronic band structure and DOS plot for cubic WO3 are shown in Figure 6.11. To understand the basic features of the electronic structure, consider the MO diagram for a d cation in octahedral coordination (see Section 6.4.2). 

---