Partially occupied bands

In the case of a partially occupied band, two cases can occnr 3) n  

Every electron is assigned a positive mass value  

It is then important to count the electrons with positive mass value and those with negative mass value. 

To deal with this difficulty, we will introduce the concept of electron holes. 

To do so, let us go back to the general expression for the current I, which can be written, if we take into account the available locations in the band  

---

The Kronig-Peimey solution illustrates that, for periodic systems, gaps ean exist between bands of energy states. As for the ease of a free eleetron gas, eaeh band ean hold 2N eleetrons where N is the number of wells present. In one dimension, tliis implies that if a well eontains an odd number, one will have partially occupied bands. If one has an even number of eleetrons per well, one will have fully occupied energy bands. This distinetion between odd and even numbers of eleetrons per eell is of fiindamental importanee. The Kronig-Penney model implies that erystals with an odd number of eleetrons per unit eell are always metallie whereas an even number of eleetrons per unit eell implies an... 

Fig. 2-6. Electron occupation in energy bands classified into (a) metals, (b) insulators, and (c) semiconductors FOB = fully occupied band FOB = partially occupied band CB = conduction band VB = valence band.

The electrical conductivity of TTF TCNQ is of the order of 10 S m at room temperature and increases with decreasing temperature until around 80 K when the conductivity drops as the temperature is lowered. TCNQ is a good electron acceptor and, for example, accepts electrons from alkali metal atoms to form ionic salts. In TTF-TCNQ, the columns of each type of molecule interact to form delocalised orbitals. Some electrons from the highest energy filled band of TTF move across to partly fill a band of TCNQ, so that both types of columns have partially occupied bands. The number of electrons transferred corresponds to about 0.69 electrons per molecule. This partial transfer only occurs with molecules such as tetrathiafulvalene whose electron donor ability is neither too small nor too large. With poor electron donors, no charge transfer... 

The main shortcoming of the cluster approach consists of the scission of the chemical bonds between terminal atoms of a cluster and the rest of a lattice. As a result, so-called dangling bonds occur at the terminal atoms of a cluster, artificial electron surface states appear in the partially occupied band, and the charge distribution is disturbed. A cluster in this case possesses too many surface atoms. Unfortunately, to obtain a better surface/bulk ratio, one should consider such large clusters that the approach becomes useless. 

In Chapter 21, we described the chemistry of the first row d-block metal(II) oxides, but said little about their electrical conductivity properties. The oxides TiO, VO, MnO, FeO, CoO and NiO adopt NaCl lattices but are non-stoichio-metric, being metal-deficient as exemplified for TiO and FeO in Section 27.2. In TiO and VO, there is overlap of the metal t2g orbitals giving rise to a partially occupied band (Figure 27.4) and, as a result, TiO and VO are electrically conducting. In contrast, MnO is an insulator at 298 K... 

While the pDOS plots of Bi and 02 indicate that the bonding between Bi and 02 is rather ionic, the sliort in-plane distance between Bi and 02 makes the in-plane hybridization of tlie Bi Px,y 02 Px,y orbitals much stronger than the out-of-plane Bi-02 hybridization. The in-plane Bi-02 interactions result in a broad-and also partially occupied-band (- 4 eV wide). 

When retaining the hypothesis of a partially occupied band of n electrons occupying N energy levels, we encounter the problem of accessing the distribution of electrons in the different levels according to temperature. 

Band filling determines the electrical properties of conventional materials. In a semiconductor E is narrow and thermal excitation of electrons from the valence band to the conduction band gives rise to conductivity. If the band gap is too wide and electrons cannot be excited thermally at room temperature across the gap, the material is an insulator. The high conductivity of metals comes from a partially occupied band or a zero band gap. 

The distortion of a regular one-dimensional structure with a partially occupied band leading to bond alternation, and to dimerization or oligomerization. The degree of oligomerization, A, depends on the electronic population of the conduction band indicated by the wave vector of the Fermi level, kf ... 

---