Fermi level and

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.)...

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.

A semiconductor can be described as a material with a Fermi energy, which typically is located within the energy gap region at any temperature. If a semiconductor is brought into electrical contact with a metal, either an ohmic or a rectifying Schouky contact is formed at the interface. The nature of the contact is determined by the workfunction, (the energetic difference between the Fermi level and the vacuum level), of the semiconductor relative to the mclal (if interface effects are neglected - see below) 47J. 

I. Riess, and C.G. Vayenas, Fermi level and potential distribution in solid electrolyte cells with and without ion spillover, Solid State Ionics, in press (2001). 

For the case of Ag it was found that both anodic and cathodic polarization lead to the creation of small insulated particles on the YSZ surface. Both the Fermi level and the work function of the insulated Ag particles was found to change with Ag electrode overpotential but the changes are smaller than on the continuous Ag film.24... 

The significant point is that PEEM, as clearly presented in Figures 5.45 to 5.47, has shown conclusively that follows reversibly the applied potential and has provided the basis for space-and time-resolved ion spillover studies of electrochemical promotion. It has also shown that the Fermi level and work function of the solid electrolyte in the vicinity of the metal electrode follows the Fermi level and work function of the metal electrode, which is an important point as analyzed in Chapter 7. 

H. Reiss, The Fermi level and the Redox potential, J. Phys. Chem. 89, 3783-3791... 

These two groups of excited carriers are not in equilibrium with each other. Each of them corresponds to a particular value of electrochemical potential we shall call these values pf and Often, these levels are called the quasi-Fermi levels of excited electrons and holes. The quasilevel of the electrons is located between the (dark) Fermi level and the bottom of the conduction band, and the quasilevel of the holes is located between the Fermi level and the top of the valence band. The higher the relative concentration of excited carriers, the closer to the corresponding band will be the quasilevel. In n-type semiconductors, where the concentration of elec-ttons in the conduction band is high even without illumination, the quasilevel of the excited electrons is just slightly above the Fermi level, while the quasilevel of the excited holes, p , is located considerably lower than the Fermi level. 

The occupation of energy level 8 depends on its position with respect to the Fermi level and should be taken into account in calculation of the transition probability. The number of electrons within a given energy interval de is equal to p(8)/(8)d8. 

At the contact of two electronic conductors (metals or semiconductors— see Fig. 3.3), equilibrium is attained when the Fermi levels (and thus the electrochemical potentials of the electrons) are identical in both phases. The chemical potentials of electrons in metals and semiconductors are constant, as the number of electrons is practically constant (the charge of the phase is the result of a negligible excess of electrons or holes, which is incomparably smaller than the total number of electrons present in the phase). The values of chemical potentials of electrons in various substances are of course different and thus the Galvani potential differences between various metals and semiconductors in contact are non-zero, which follows from Eq. (3.1.6). According to Eq. (3.1.2) the electrochemical potential of an electron in... 

The electronic conductivity of metal oxides varies from values typical for insulators up to those for semiconductors and metals. Simple classification of solid electronic conductors is possible in terms of the band model, i.e. according to the relative positions of the Fermi level and the conduction/valence bands (see Section 2.4.1). 

In addition to the stoichiometry of the anodic oxide the knowledge about electronic and band structure properties is of importance for the understanding of electrochemical reactions and in situ optical data. As has been described above, valence band spectroscopy, preferably performed using UPS, provides information about the distribution of the density of electronic states close to the Fermi level and about the position of the valence band with respect to the Fermi level in the case of semiconductors. The UPS data for an anodic oxide film on a gold electrode in Fig. 17 clearly proves the semiconducting properties of the oxide with a band gap of roughly 1.6 eV (assuming n-type behaviour). 

Since the energy of electrons in a material is specified by the Fermi level, ep, the flow of electrons across an interface must likewise depend on the relative Fermi levels of the materials in contact. Redox properties are therefore predicted to be a function of the Fermi energy and one anticipates a simple relationship between the Fermi level and redox potential. In fact, the Fermi level is the same as the chemical potential of an electron. Clearly when dealing with charged particles, the local energy levels e are increased by qV, where q is the charge on the particle and V is the local electrostatic potential. The e, should therefore be replaced by e,- + qV and so... 

The band structure that appears as a consequence of the periodic potential provides a logical explanation of the different conductivities of electrons in solids. It is a simple case of how the energy bands are structured and arranged with respect to the Fermi level. In general, for any solid there is a set of energy bands, each separated from the next by an energy gap. The top of this set of bands (the valence band) intersects the Fermi level and will be either full of electrons, partially filled, or empty. 

At zero temperature the electrons in a solid occupy the lowest energy levels compatible with the Pauli exclusion principle. The highest energy level occupied at T = 0 is the Fermi level, Ep. For metals the Fermi level and the electrochemical potential are identical at T = 0, since any electron that is added to the system must occupy the Fermi level. At finite temperatures Ep and the electrochemical potential p of the electrons differ by terms of the order of (kT)2, which are typically... 

The last approximation is valid if Ec — Ep kT (i.e., if the band edge is at least a few kT above the Fermi level), and the Fermi-Dirac distribution /(e) can be replaced by the Boltzmann distribution. Similarly, the concentration of holes in the valence band is ... 

If the electronic properties of the semiconductor - the Fermi level, the positions of the valence and the conduction band, and the flat-band potential - and those of the redox couple - Fermi level and energy of reorganization - are known, the Gerischer diagram can be constructed, and the overlap of the two distribution functions Wox and Wred with the bands can be calculated. 

The opposite occurs for atoms with a high electron affinity that is on the order of the metal work function or higher. Here the broadened level 2 falls partly below the Fermi level and becomes partially occupied (Fig. A. 10c). In this case the adatom is negatively charged. Examples are the adsorption of electronegative species such as F and Cl. Table A.3 gives ionization potentials and electron affinities of some catalytically relevant atoms. 

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