Fermi-level mechanism

It is well established, that the poor selectivity of tin-oxide sensors can partly be overcome by adding catalysts to the sensitive layer. Most common additives are noble metals like gold (Au), platinum (Pt) or palladium (Pd). They can be mixed with the tin oxide during paste formation before deposition. The influence of dopants on the gas sensor response is still subject to debates. The two most established mechanisms are the spill-over and the Fermi-level mechanism [82]. 

Fig. 2.4. Effects of noble-metal catalyst doping on the tin oxide (a) SpiU-over mechanism, (b) Fermi-level mechanism...

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... 

MetaUic behavior is observed for those soHds that have partially filled bands (Fig. lb), that is, for materials that have their Fermi level within a band. Since the energy bands are delocalized throughout the crystal, electrons in partially filled bands are free to move in the presence of an electric field, and large conductivity results. Conduction in metals shows a decrease in conductivity at higher temperatures, since scattering mechanisms (lattice phonons, etc) are frozen out at lower temperatures, but become more important as the temperature is raised. 

Figure 6.14d shows the electron donation interaction (electrons are transferred from the initially fully occupied 5a molecular orbitals to the Fermi level of the metal, thus this is an electron donation interaction). Blyholder was first to discuss that CO chemisorption on transition metal involves both donation and backdonation of electrons.4 We now know both experimentally7 and theoretically96,98 that the electron backdonation mechanism is usually predominant, so that CO behaves on most transition metal surfaces as an overall electron acceptor. 

The dependence of g0 on the position of the Fermi level at the surface of the catalyst is presented in Fig. 11 by a heavy curve, in accordance with (80) and (7). We see that within the framework of the mechanism considered above the oxidation of CO falls into the category of acceptor reactions (i.e., it is retarded as the Fermi level is lowered). 

The other mechanism involves atomic-size roughness (i.e., single adatoms or small adatom clusters), and is caused by electronic transitions between the metal and the adsorbate. One of the possible mechanisms, photoassisted metal to adsorbate charge transfer, is illustrated in Fig. 15.4. It depends on the presence of a vacant, broadened adsorbate orbital above the Fermi level of the metal (cf. Chapter 3). In this process the incident photon of frequency cjq excites an electron in the metal, which subsequently undergoes a virtual transition to the adsorbate orbital, where it excites a molecular vibration of frequency lj. When the electron returns to the Fermi level of the metal, a photon of frequency (u>o — us) is emitted. The presence of the metal adatoms enhances the metal-adsorbate interaction, and hence increases the cross... 

To conclude, even if there exist several processes that affect the vibrational line shape it seems probable that when most of them have been sorted out and with the good agreement between theory and experiment, the lifetime broadening for a chemisorbed CO molecule is of the order of a few cm, corresponding to a lifetime of a few ps. The main vibrational energy relaxation mechanism is creation of electron-hole pairs caused by the local charge oscillations between the metal and the 2n molecular resonance crossing the Fermi level. 

As mentioned in Sec. 1.3, the electrochemical potential of electrons in condensed phases corresponds to the Fermi level of electrons in the phases. There are two possible cases of electron ensembles in condensed phases one to which the band model is applicable (in the state of degenera< where the wave functions of electrons overlap), and the other to which the band model cannot apply (in the state of nondegeneracy where no overlap of electron wave functions occurs). In the former case electrons or holes are allowed to move in the bands, while in the latter case electrons are assumed to be individual particles rather than waves and move in accord with a thermal hopping mechanism between the a4jacent sites of localized electron levels. 

Figures 8-16 and 8-17 show the state density ZXe) and the exchange reaction current io( ) as functions of electron energy level in two different cases of the transfer reaction of redox electrons in equilibrium. In one case in which the Fermi level of redox electrons cnxEDax) is close to the conduction band edge (Fig. 8-16), the conduction band mechanism predominates over the valence band mechanism in reaction equilibrium because the Fermi level of electrode ensa (= nREDOK)) at the interface, which is also dose to the conduction band edge, generates a higher concentration of interfadal electrons in the conduction band than interfadal holes in the valence band. In the other case in which the Fermi level of redox electrons is dose to the valence band edge (Fig. 8-17), the valence band mechanism predominates over the conduction band mechanism because the valence band holes cue much more concentrated than the conduction band electrons at the electrode interface.

From these illustrations it follows, in general, that Ihe transfer reaction of redox electrons at semiconductor electrodes occurs via the conduction band mechanism if its equilibrium potential is relatively low (high in the Fermi level of redox electrons) whereas, the transfer reaction of redox electrons proceeds via the valence band mechanism if the equilibriiun redox potential is high (low in the Fermi level of redox electrons). 

A further increase in anodic polarization lowers still further the Fermi level ersc)(ti) which gradually approaches the valence band edge Cy at the electrode interface as shown in Fig. 8-21. As the anodic polarization increases, the concentration of interfacial holes in the valence band increases, thus causing the anodic electron transfer to change from the conduction band mechanism to the valence band mechanism. 

It follows that, when Ptk e, is greater than (e -Ev), the rate of capture of holes is greater than the rate of release of electrons (Vb>Vt). Namely, if the Fermi level ej at the surface is lower than the middle, (ej + Ev)/2, of the band gap (the p-type surface) and if p.b is close to 0.5, the valence band mechanism of Eqn. 9-24b will probably predominate over the conduction band mechanism of Eqn. 9-24a. As the band gap e, of semiconductor electrode increases, the valence band mechanism become more predominant. 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

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