Band structure diagrams

A comparison of the band structure diagram and these two measurements shows that experimentally the main measured intensity is constrained to a few of the bands present. In the first Brillouin zone the ct, band is found to be occupied, in the second zone 02. No sign of o, or the % band is found for the T M measurement. For the A-L measurement the same bands as for the T-M measurement contribute but in addition the n band is observed, mainly in the first Brillouin zone. These experiments are a beautiful, direct observation of the nodal plane of the % electrons in momentum space. 

Fig. 5.3. Energy band-structure diagram (in eV) of Ni/ZnO support and pre-(post-)chemisorbed hydrogen adatom level at e0(e ). VB (shaded) and CB of ZnO are of width 6. Fermi level (e/), which coincides with lower edge of CB, is taken as zero of energy. 6-layer Ni film has 6 localized levels lying between band edges (dashed lines), which just overlap ZnO energy gap. Reprinted from Davison et al (1988) with permission from Elsevier.

Figure 4.7 (a) The band structure diagram of Si near the gap energy (reproduced with per-... 

Indirect transitions are much weaker thau direct trausitious, because the latter do uot require the participation of photons. However, many indirect-gap materials play an important role in technological applications, as is the case of silicon (band structure diagram iu Figure 4.7(a)) or germanium (baud structure diagram shown later, in Figure 4.11). Hereafter, we will deal with the spectral shape expected for both direct and indirect transitions. 

FIGU RE 6.16 (a) Band structure diagram for KC8 calculated using the self-consistent LCAO method. Lowest... 

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. 

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. 

In two dimensions, these functions are surfaces. Eigure 5.3 shows the band structure diagram of graphene. 

Figure 2.3 (a) Band structure diagram showing the same processes as in Fig. 2.1a. The field acceleration...

Fig. 4.4 A portion of a schematic band structure diagram showing the energy as a fimction of k in a particular direction o f A -space. for the valence and conduction bands. The minimum energy difference Eg is the band gap. /i is the electron chemical potential.

Figure 7. Band structure diagram for a iinear carbon chain with C - C of 220 pm and one carbon atom per unit ceii. The crystai orbitais are sketched schematicaiiy at T and Z.

The 4f states of the Tm " ions are clearly seen in the band gap. Because Tm " ions have 13 4f electrons, spin-polarized calculations were performed to distinguish between the spin-up and spin-down states. The band-structure diagrams from Fig. 5.5 allow for the following estimation of the position of the Tm " 4f ground state in the band gap above the top of the host s valence band 1.05 and 2.91 eV (spin-up and spin-down states in CaCl2, respectively), 1.37 and 3.20 eV (spin-up and spin-down states in SrCl2, respectively), and 1.39 and 3.12 eV (spin-up and spin-down states in BaCl2, respectively). 

Fig. 5.5 Calculated band-structure diagrams for Tm -doped MCL [M = Ca (a and b), Sr (c and d), and Ba (e and f)] crystals. The red two-way arrows Indicate the energy separation between the host valence and conduction bands. ALPHA and BETA represent the electronic pictures with the up and down spins, respectively, c, d Adapted from Ref. [10] by permission of John Wiley Sons Ltd...

A more complete view of the electronic structure can be obtained from quantum mechanics. All of this information is contained within the Brillouin zone, but this gives us a conceptual problem it is a complex three-dimensional shape that resides in reciprocal space. We can simplify any three-dimensional shape by cutting slices through it to create two-dimensional representations. In this way, a sphere becomes a circle and so on. We can do the same with the Brillouin zone. By slicing through certain pathways, called k vectors, which link k points (which are special positions in the three-dimensional Brillouin zone defined by the real-space crystal system) we generate the two-dimensional band structure diagrams. 

The key observation to note from the band structure diagram is that it is rather flat, with a reasonably large predicted band gap (779 nm compared to the experimentally derived 928 nm). The material is therefore an insulator, and the lack of variation in the energies of the individual bands (sometimes also referred to as... 

Sketch a band structure diagram for the crystal showing qualitatively the energies of the crystal orbitals as a function of k in the first Brillouin zone. Sketch the appearances of the COs for the cr and a bands at = 0 and nia. 

Fig. 18.3 Simplified band structure diagrams of a metal, semiconductor, and insulator. Typical values of the band gap are 0 eV in metals, 0.5-5.0 eV in semiconductors, and 5 eV or greater in insulators...

Fig. 6a. Schematic diagram relating duster states to bulk crystal states adapted from [13]. b Schematic band structure diagram cubic crystalline CdS. The lines labeUL(lli) and X (100) r er to different directions within the unit cell [12]...

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