Solids Fermi level

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... 

Fig. 5. Electrical resistance as a function of the temperature at the indicated magnetic fields for a single microbundle of carbon nanotubes. The solid line is a fit using the two-band model for graphite (see inset) with an overlap A = 3.7 meV and a Fermi level right in the middle of the overlap (after Langer et at. l9 ).

For compositions with 50 and 57% sodium Ep lies within the tail of the partial DOS of the sp-band of tin, but Ep has still not reached the region where the partial DOS of sodium is large. This yields a small DOS (pseudogap) for these cases at the Fermi level. Therefore, one gets an explanation for the minimum of the conductivity (i.e. the maximum of the resistivity) near the equimolar composition, as can be seen in Table 1. (Analogously, for solid equimolar /3-NaSn even a indirect band gap at the Fermi level was reported in literature [16].)... 

The PES measurements arc performed with reference to the Fermi level of the photoclectron spectrometer, in solid specimens, as dealt with here, by the way the spectroscopy works. Thus, in cases when the Fermi level shifts due to some chemical modifications of the sample, i.e., in the intercalation of graphite or other layered compound [16] or in the doping of conjugated polymers 1171, il is necessary to account for the change in the Fermi energy level before interpreting spec-... 

As discussed already in Chapter 2 the work function, , of a solid surface is one of the most important parameters dictating its chemisorptive and catalytic properties. The work function, (eV/atom) of a surface is the minimum energy which an electron must have to escape from the surface when the surface is electrically neutral. More precisely is defined as the energy to bring an electron from the Fermi level, EF, of the solid at a distance of a few pm outside of the surface under consideration so that image charge interactions are negligible. 

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). 

Figure 5.17. Schematic representation of a metal crystallite deposited on YSZ and of the changes induced in its electronic properties upon polarizing the catalyst-solid electrolyte interface and changing the Fermi level (or electrochemical potential of electrons) from an initial value p to a new value p -eri30 31 Reprinted with permission from Elsevier Science.

It must be emphasized that Equations (5.24) and (5.25) stem from the definitions of Fermi level, work function and Volta potential and are generally valid for any electrochemical cell, solid state or aqueous. We can now compare these equations with the corresponding experimental equations (5.18) and (5.19) found to hold, under rather broad temperature, gaseous composition and overpotential conditions (Figs. 5.8 to 5.16), in solid state electrochemistry ... 

Figure 5.45 shows a Pt electrode (light) deposited on YSZ (dark). There are three circular areas of bare YSZ connected via very narrow bare YSZ channels. The rest of the surface is Pt. Note that, as will be discussed in Chapter 7, the Fermi levels of the Pt film and of the YSZ solid electrolyte in the vicinity of the Pt film are equal. The YSZ, however, appears in the PEEM images much darker than the Pt film since YSZ has a negligible density of states at its Fermi level in comparison to a metal like Pt. 

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. 

It will also be shown that the absolute electrode potential is not a property of the electrode but is a property of the electrolyte, aqueous or solid, and of the gaseous composition. It expresses the energy of solvation of an electron at the Fermi level of the electrolyte. As such it is a very important property of the electrolyte or mixed conductor. Since several solid electrolytes or mixed conductors based on ZrC>2, CeC>2 or TiC>2 are used as conventional catalyst supports in commercial dispersed catalysts, it follows that the concept of absolute potential is a very important one not only for further enhancing and quantifying our understanding of electrochemical promotion (NEMCA) but also for understanding the effect of metal-support interaction on commercial supported catalysts. 

This, at first perhaps surprising fact, is important to remember as the same situation arises in solid state electrochemistry. To understand its validity it suffices to remember that the definition of the reference (zero) energy level of electrons for the she scale is simply the state of an electron at the Fermi level of any metal in equilibrium with an aqueous solution of pH=0 and pH2=l atm at 25°C. 

It is thus clear from the previous discussion that the absolute electrode potential is not a property of the electrode material (as it does not depend on electrode material) but is a property of the solid electrolyte and of the gas composition. To the extent that equilibrium is established at the metal-solid electrolyte interface the Fermi levels in the two materials are equal (Fig. 7.10) and thus eU 2 (abs) also expresses the energy of transfering an electron from the Fermi level of the YSZ solid electrolyte, in equilibrium with po2=l atm, to a point outside the electrolyte surface. It thus also expresses the energy of solvation of an electron from vacuum to the Fermi level of the solid electrolyte. 

Equation (7.32) underlines the pinning of the Fermi levels of metal electrodes with the solid electrolyte and reminds the fact that the absolute electrode potential is a property of the solid electrolyte and of the gaseous composition but not of the electrode material.21... 

Consequently the absolute potential is a material property which can be used to characterize solid electrolyte materials, several of which, as discussed in Chapter 11, are used increasingly in recent years as high surface area catalyst supports. This in turn implies that the Fermi level of dispersed metal catalysts supported on such carriers will be pinned to the Fermi level (or absolute potential) of the carrier (support). As discussed in Chapter 11 this is intimately related to the effect of metal-support interactions, which is of central importance in heterogeneous catalysis. 

By comparing Figure 11.9 and the characteristic Po2(Uwr) rate breaks of the inset of Fig. 11.9 one can assign to each support an equivalent potential Uwr value (Fig. 11.10). These values are plotted in Figure 11.11 vs the actual work function G>° measured via the Kelvin probe technique for the supports at po2-l atm and T=400°C. The measuring principle utilizing a Kelvin probe and the pinning of the Fermi levels of the support and of metal electrodes in contact with it has been discussed already in Chapter 7 in conjunction with the absolute potential scale of solid state electrochemistry.37... 

Fermi level of electrons and absolute potential, 346 distribution in a solid electrolyte cell, 219, 357... 

The work function is the minimum energy needed to remove an electron from a solid and take it infinitely far away at zero potential energy. The weakest bound electrons in a solid are the electrons at the Fermi level. All sp and d bands are filled... 

For a material to be a good conductor it must be possible to excite an electron from the valence band (the states below the Fermi level) to the conduction band (an empty state above the Fermi level) in which it can move freely through the solid. The Pauli principle forbids this in a state below the Fermi level, where all states are occupied. In the free-electron metal of Fig. 6.14 there will be plenty of electrons in the conduction band at any nonzero temperature - just as there will be holes in the valence band - that can undertake the transport necessary for conduction. This is the case for metals such as sodium, potassium, calcium, magnesium and aluminium. 

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

In solid-state physics, the electrochemical potential of the electron pe(a) is mostly replaced by the equivalent energy of the Fermi level eF. While the electrochemical potential is usually related to one mole of particles, the Fermi energy is related to a single electron, so that... 

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). 

---