Fermi redox

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

H. Gerischer, and W. Ekardt, Fermi levels in electrolytes and the absolute scale of redox potentials, Appl. Phys. Lett. 43(4), 393-395 (1983). 

Singh P, Singh R, Gale R, Rajeshwar K, DuBow J (1980) Surface charge and specific ion adsorption effects in photoelectrochemical devices. J Appl Phys 51 6286-6291 Bard AJ, Bocarsly AB, Pan ERF, Walton EG, Wrighton MS (1980) The concept of Fermi level pinning at semiconductor/liquid junctions. Consequences for energy conversion efficiency and selection of useful solution redox couples in solar devices. J Am Chem Soc 102 3671-3677... 

It follows from the Franck-Condon principle that in electrochemical redox reactions at metal electrodes, practically only the electrons residing at the highest occupied level of the metal s valence band are involved (i.e., the electrons at the Fermi level). At semiconductor electrodes, the electrons from the bottom of the condnc-tion band or holes from the top of the valence band are involved in the reactions. Under equilibrium conditions, the electrochemical potential of these carriers is eqnal to the electrochemical potential of the electrons in the solution. Hence, mntnal exchange of electrons (an exchange cnrrent) is realized between levels having the same energies. 

Because of the excess holes with an energy lower than the Fermi level that are present at the n-type semiconductor surface in contact with the solution, electron ttansitions from the solution to the semiconductor electrode are facilitated ( egress of holes from the electrode to the reacting species ), and anodic photocurrents arise. Such currents do not arise merely from an acceleration of reactions which, at the particular potential, will also occur in the dark. According to Eq. (29.6), the electrochemical potential, corresponds to a more positive value of electrode potential (E ) than that which actually exists (E). Hence, anodic reactions can occur at the electrode even with redox systems having an equilibrium potential more positive than E (between E and E ) (i.e., reactions that are prohibited in the dark). 

The photoelectrolysis of H2O can be performed in cells being very similar to those applied for the production of electricity. They differ only insofar as no additional redox couple is used in a photoelectrolysis cell. The energy scheme of corresponding systems, semiconductor/liquid/Pt, is illustrated in Fig. 9, the upper scheme for an n-type, the lower for a p-type electrode. In the case of an n-type electrode the hole created by light excitation must react with H2O resulting in 02-formation whereas at the counter electrode H2 is produced. The electrolyte can be described by two redox potentials, E°(H20/H2) and E (H20/02) which differ by 1.23 eV. At equilibrium (left side of Fig. 9) the electrochemical potential (Fermi level) is constant in the whole system and it occurs in the electrolyte somewhere between the two standard energies E°(H20/H2) and E°(H20/02). The exact position depends on the relative concentrations of H2 and O2. Illuminating the n-type electrode the electrons are driven toward the bulk of the semiconductor and reach the counter electrode via the external circuit at which they are consumed for Hj-evolution whereas the holes are dir tly... 

The potential which controls the photoelectrochemical reaction is generally not the photopotential defined by Eqs (5.10.20) and (5.10.21) (except for the very special case where the values of v, REdox and the initial Fermi energy of the counterelectrode are equal). The energy which drives the photoelectrochemical reaction, eR can be expressed, for example, for an n-semiconductor electrode as... 

The conclusions from these considerations are that semiconductor photoelectrodes can be used to effect either reductions (p-type semiconductors) or oxidations (n-type semiconductors) in an uphill fashion. The extent to which reaction can be driven uphill, Ey, is no greater than Eg, but may be lower than Eg owing to surface states between Eqb and Eye or to an Inappropriate value of Ere(jox. Both Eg and Epg are properties that depend on the semiconductor bulk and surface properties. Interestingly, Ey can be independent of Ere(jox meaning that the choice of Ere(jox and the associated redox reagents can be made on the basis of factors other than theoretical efficiency, for a given semiconductor. Thus, the important reduction processes represented by the half-reactions (3)-(5) could, in principle, be effected with the same efficiency at a Fermi level pinned (or... 

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... 

To show that the Fermi level can be interpreted as the redox potential, consider a redox equilibrium such as... 

In this chapter we introduce and discuss a number of concepts that are commonly used in the electrochemical literature and in the remainder of this book. In particular we will illuminate the relation of electrochemical concepts to those used in related disciplines. Electrochemistry has much in common with surface science, which is the study of solid surfaces in contact with a gas phase or, more commonly, with ultra-high vacuum (uhv). A number of surface science techniques has been applied to electrochemical interfaces with great success. Conversely, surface scientists have become attracted to electrochemistry because the electrode charge (or equivalently the potential) is a useful variable which cannot be well controlled for surfaces in uhv. This has led to a laudable attempt to use similar terminologies for these two related sciences, and to introduce the concepts of the absolute scale of electrochemical potentials and the Fermi level of a redox reaction into electrochemistry. Unfortunately, there is some confusion of these terms in the literature, even though they are quite simple. 

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. 

Figure 2.4 Shift of the metal Fermi level on application of an overpotential, which is the equilibrium potential fa of the redox couple, is ...

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... 

Figure 7.6 Gerischer diagram for a redox reaction at an n-type semiconductor (a) at equilibrium the Fermi levels of the semiconductor and of the redox couple are equal (b) after application of an anodic overpotential.

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 ... 

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. 

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