Fermi level of redox electron

Fig. 2-39. Gaussian normal distri bution of the probabili density of redox electron levels due to thermal fluctuation of hydrate structures epd)Bx>X) = standard Fermi level of redox electrons.

Fig. 2-40. Distribution of electron state density of hydrated redox particles (a) oxidant concentration JVox equal to reductant concentrationNRED. (b) oxidant concentration iVox greater than reductant concentration NgEo cnsEDox) = Fermi level of redox electrons.

Fig. 4-17. Electronic electrode in equilibrium of electron transfer OX = hydrated oxidant particles RED = hydrated reductant partides FWEDQx, s) = Fermi level of redox electrons in hydrated redox partides in solution S p. = electrochemical potential of electrons.

Figure 5-64 shows the band edge potential for compound semiconductor electrodes in aqueous solutions, in which the standard redox potentials (the Fermi levels) of some hydrated redox particles are also shown on the right hand side. In studying reaction kinetics of redox electron transfer at semiconductor electrodes, it is important to find the relationship between the band edge level (the band edge potential) and the Fermi level of redox electrons (the redox potential) as is described in Chap. 8. 

Fig. 8-2. Electron state draisity in a metal electrode and in hydrated redox particles on both sides of an electrode interface in equilibrium with redox electron transfer I>m = state density of electrons in metal electrode Oo Dhbd)=state density of redox electrons in hydrated oxidant (reductant) particle cfcredox) = Fermi level of redox electrons ...

Equation 2-51 gives the Fermi level of redox electrons as a function of the concentration of redox particles as shown in Eqn. 8-12 ... 

At the Fermi level of redox electrons the whole state density DraDosKeroasDox)) of redox particles is half occupied by electrons that provide the occupied state density I>KED(er( U Dox)) of reductant particles, and the remaining half is vacant for electrons in the imoccupied state density -DoxCennEDox)) of oxidant particles. Consequently, to a first approximation, the state densities Dred(e) and i ox(e) at enei level e near the Fermi level of redox electrons can be derived, respectively, to produce Eqns. 8-15 and 8-16 [Gerischer, I960] ... 

Fig. 8-3. Electron state density in hydrated reductant and oxidant particles near the Fermi level of redox electrons I>redox = 1)red -I>ox= electron state density in redox particles Dpcredox) 1)f(bed)l nox) = electron state density in redox particles at the Fermi level of redox electrons.

In transfer equilibrium of redox electrons, the Fermi level of the electrode crm) equals the Fermi level of the redox particles crredox). at the electrode interface. Hence, with the standard Fermi level, of redox electrons, we obtain the... 

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). 

Fig. 8-27. Polarization curves for transfer of redox electrons at n-type and p-type semiconductor electrodes solid curve near Egaxa = reaction with the Fermi level of redox electrons dose to the valence band edge dashed curve near F redok = reaction with the Fermi level of redox electrons dose to the conduction band edge dot-dash curve (FLP)= reaction in the state of Fermi level pinning.

Complexation therefore raises the standard Fermi level of redox electrons ep(KEDcs)> provided that the affinity of ligand coordination is greater with the oxidant particle than with the reductant particle (- dGox > - dG c) whereas, the complexation lowers the standard Fermi level of redox electrons Ef(redox)> provided that the affinity of ligand coordination is smaller with the oxidant particle than with the reductant particle (- dGox < - dG o)- With a shift of the standard Fermi level of redox electrons due to complexation, the most probable electron levels esED and cox of the redox particles are also shifted in the same direction. 

The kinetic treatment for the electron transfer of ligand-coordinated redox particles described in Sec. 8.4.1 may, in principal, apply also to the electron transfer of adsorbed redox particles (inner-sphere electron transfer). The contact adsorption of redox particles on metal electrodes requires the dehydration of hydrated redox particles and hence inevitably shifts the standard Fermi level of redox electrons from in the hydrated state to in the adsorbed state. This shift of the Fermi level of redox electrons due to the contact adsorption of redox particles is expressed in Eqn. 8-83 similarly to Eqn. 8-79 for the complexation of redox particles (ligand coordination) ... 

FIGURE 22.2 Electron energy levels for a standard pair of hydrated redox particles and for an intrinsic semiconductor ered = the most probable electron level of oxidant, eox = the most probable electron level of reductant, 8p(redox) = standard Fermi level of redox electrons, 8p = Fermi level of an intrinsic semiconductor, v = valence band edge level, and c = conduction band edge level. 

It is characteristic of metal electrodes that the reaction current of redox electron transfer, under the anodic and cathodic polarization conditions, occurs mostly at the Fermi level of metal electrodes rather than at the Fermi level of redox particles. In contrast to metal electrodes, as is discussed in Sec. 8.2, semiconductor electrodes exhibit no electron transfer current at the Fermi level of the electrodes. 

In the equilibriiun of interfacial redox reactions of the adsorbed protons and hydrogens, the Fermi level of semiconductor electrons at the electrode interface equals the Fermi level e p(h /h) of interfacial redox electrons in the adsorbed protons and hydrogens. The Fermi level e gc) th interface of semiconductor electrode depends on the potential /l< )sc of the space charge layer as shown in Eqn. 9-66 ... 

Figure 10-3 juxtaposes the Fermi levels of the following redox reactions in aqueous solutions and the quasi-Fermi levels of interfacial electrons and holes in an n-type semiconductor electrode erhjo/Hj) of the hydrogen redox reaction F(0a/H20) of the oxj en redox reaction ersc) of the n- q)e semiconductor and... 

Fig. 10-36. Polarization curves for a redox reaction at an n-type anode and at a p-lype cathode in a photovoltaic cell solid curve n-SC = anodic current at photoexdted n-type anode (Fermi level versus current curve) dashed curve p-SC = anodic current at dark p-type anode (current versus quasi-Fermi level of interfadal holes in photoexdted n-type anode) solid curve p-SC = cathodic current at photoexdted p-type cathode (Fermi level versus current curve) dashed curve n-SC = cathodic current at daric n-type cathode (current versus quasi-Fermi level of interfadal electrons in a photoexdted p- q>e cathode).

Fig. 1 A scheme of the energetics at an n-type semiconductor electrode in contact with a redox system in an electrolyte solution, (a) The situation under conditions of electronic equilibrium. The electrochemical potential of the electrons is the same in both phases, i.e. the electron Fermi-level in the semiconductor Ep.n has the same value as the Fermi-level of the electrons in the redox system fRed/Ox- (t>) Case in which the energy bands in the semiconductor are flat this situation, corresponding to maximum photovoltage, is reached under strong illumination at open circuit. Wsc is the width of the depletion layer and e(j>sc is the band-bending.

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... 

For a metal, the negative of the work function gives the position of the Fermi level with respect to the vacuum outside the metal. Similarly, the negative of the work function of an electrochemical reaction is referred to as the Fermi level Ep (redox) of this reaction, measured with respect to the vacuum in this context Fermi level is used as a synonym for electrochemical potential. If the same reference point is used for the metal s,nd the redox couple, the equilibrium condition for the redox reaction is simply Ep (metal)= Ep(redox). So the notion of a Fermi level for a redox couple is a convenient concept however, this terminology does not imply that there are free electrons in the solution which obey Fermi-Dirac statistics, a misconception sometimes found in the literature. 

In the preceding derivation we presumed that equilibrium prevails, so that the Fermi levels of the metal and of the redox couple are equal. This equilibrium can be disturbed by the application of an external electrode potential fa, which lowers the electronic energies in the metal, and in particular the Fermi level, by an amount — eo 7, where r) — (f> — fa is called the overpotential (see Fig. 2.4). Thus the application of an overpotential leads to a difference —eor] in the Fermi levels of the metal and the solution. However, as the equilibrium is... 

Typically the contributions of the two bands to the current are of rather unequal magnitude, and one of them dominates the current. Unless the electronic densities of states of the two bands differ greatly, the major part of the current will come from the band that is closer to the Fermi level of the redox system (see Fig. 7.6). The relative magnitudes of the current densities at vanishing overpotential can be estimated from the explicit expressions for the distribution functions Wled and Wox ... 

In the fluctuation band of electron energy of hydrated redox particles, the donor band of the reductant is an occupied band, and the acceptor band of the oxidant is a vacant band. The level erotsDcno at which the donor state density equals the acceptor state density (Aai/e) = Dox(e)) is called the Fermi level of the redox electron by analogy with the Fermi level e, of metal electrons [Gerischer, 1961]. From Eqns. 2—48 and 2—49 with f BED(e) =-DoxCe), we obtain the Fermi level Tiixxox.) (the redox electron level) as shown in Eqn. 2-51 ... 

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