The Fermi level

The energy level of the highest occupied orbital in a metal at absolute zero is called the Fermi level. At this temperature, the electronic configuration predicted by the aufbau principle appertains and so, in Li for example, the Fermi level lies exactly at the centre of the half-filled band. For other metals, the Fermi level lies at or near the centre of the band. At temperatures above OK, electrons thermally populate MOs just above the Fermi level, and some energy levels just below it remain unoccupied. In the case of a metal, the thermal populations of different energy states cannot be described in terms of a Boltzmann distribution, but are instead given by the Fermi-Dirac distribution  

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While field ion microscopy has provided an effective means to visualize surface atoms and adsorbates, field emission is the preferred technique for measurement of the energetic properties of the surface. The effect of an applied field on the rate of electron emission was described by Fowler and Nordheim [65] and is shown schematically in Fig. Vlll 5. In the absence of a field, a barrier corresponding to the thermionic work function, prevents electrons from escaping from the Fermi level. An applied field, reduces this barrier to 4> - F, where the potential V decreases linearly with distance according to V = xF. Quantum-mechanical tunneling is now possible through this finite barrier, and the solufion for an electron in a finite potential box gives... 

Fig. VIII-5. Schematic potential energy diagram for electrons in a metal with and without an applied field , work function Ep, depth of the Fermi level. (From Ref. 62.)...

Fig. XVIII-16. A four-electron two-orbital interaction that a) has no net bonding in the free molecule but can be bonding to a metal surface if (b) the Fermi level is below the antibonding level. In the lower part of the figure, a zero-electron two-orbital situation (c) has no bonding but there can be bonding to a metal surface as in (d) if the Fermi level is above the bonding level. (From Ref. 160.)...

Fig. XVIII-18. Interaction of the a and n molecular orbitals with the Pt d band Ef is the Fermi level. (From Ref. 172.)...

Fig. XVIII-19. Band bending with a negative charge on the surface states Eu, E/, and Ec are the energies of the valance band, the Fermi level, and the conduction level, respectively. (From Ref. 186.)...

One important question is how many orbitals are available at any given energy level. This is shown using a density of states (DOS) diagram as in Figure 34.2. It is typical to include the Fermi level as denoted by the dotted line in this figure. A material with a half-filled energy band is a conductor, but it may be a... 

A semiconductor laser takes advantage of the properties of a junction between a p-type and an n-type semiconductor made from the same host material. Such an n-p combination is called a semiconductor diode. Doping concentrations are quite high and, as a result, the conduction and valence band energies of the host are shifted in the two semiconductors, as shown in Figure 9.10(a). Bands are filled up to the Fermi level with energy E. ... 

Fig. 1. Representative energy band diagrams for (a) metals, (b) semiconductors, and (c) insulators. The dashed line represents the Fermi Level, and the shaded areas represent filled states of the bands. denotes the band gap of the material.

Instead of plotting the electron distribution function in a band energy level diagram, it is convenient to indicate the Fermi level. For instance, it is easy to see that in -type semiconductors the Fermi level Hes near the valence band. 

Fig. 2. Representation of the band edges in a semiconductor p—n junction where shallow donor, acceptor energies, and the Fermi level are labeled Ejy E, and E respectively, (a) Without external bias is the built-in potential of the p—n junction (b) under an appHed forward voltage F. ...

Flowever, when the metal can be detected directly (mainly Pt), it is possible to relate the form of the NMR spectmm to the dispersion of the metal and to calculate the electron density of states at the Fermi level. 

By contrast, in EELS the characteristic edge shapes are derived from the excitation of discrete inner shell levels into states above the Fermi level (Figure lb) and reflect the empty density of states above f for each atomic species. The overall... 

If a voltage V is applied then the Fermi levels Ep are shifted against each other by an energy ex V, where e is the electrostatic charge of an electron. Because of the energy... 

The doping of Ceo with alkali metals creates carriers at the Fermi level in the tiu-derived band and decreases the electrical resistivity p of pristine solid Ceo by several orders of magnitude. As x in Ma C6o increases, the resistivity p(.-r) approaches a minimum at x = 3.0 0.05 [9, 112], corresponding to a half-filled flu-derived conduction band. Then, upon further increase in x from 3 to 6, p x) again increases, as is shown in Fig. 11 for various alkali metal dopants... 

A symmetry-imposed band degeneracy occurs for the [(/i-3/2)k bands at the Fermi level. 

Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

Fig. 8. LDF valence band structure of [9,2] chiral nanotube. The Fermi level lies at midgap at -3.3 eV. Dimensionless wavenumber coordinate k ranges from 0 to t.

The optimised interlayer distance of a concentric bilayered CNT by density-functional theory treatment was calculated to be 3.39 A [23] compared with the experimental value of 3.4 A [24]. Modification of the electronic structure (especially metallic state) due to the inner tube has been examined for two kinds of models of concentric bilayered CNT, (5, 5)-(10, 10) and (9, 0)-(18, 0), in the framework of the Huckel-type treatment [25]. The stacked layer patterns considered are illustrated in Fig. 8. It has been predicted that metallic property would not change within this stacking mode due to symmetry reason, which is almost similar to the case in the interlayer interaction of two graphene sheets [26]. Moreover, in the three-dimensional graphite, the interlayer distance of which is 3.35 A [27], there is only a slight overlapping (0.03-0.04 eV) of the HO and the LU bands at the Fermi level of a sheet of graphite plane [28,29],... 

This restriction, however, could be circumvented by the doped CNT with either Lewis acid or base [32-36], since such doping, even to semiconductive CNT could enhance the density of states at the Fermi level as well as bring about the metallic property. Appearance of metallic conductivity in helical CNT by such doping process would be of interest in that it could make molecular solenoid of nanometer size [37]. 

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