The solution Fermi level

The concept of the solution Fermi level is very useful in photoelectrochemistry, althongh it strikes some as strange at first, because the term Fermi level was first introduced and defined for an electronically conducting phase, such as a metal or semiconductor, which contains free electrons. In this context, the Fermi level is defined as the energy, measured with respect any convenient reference level, for which the probability that an electronic energy level is occupied is one-half. The most convenient reference level to use in photoelectrochemistry is the local vacuum level of the solution. The Fermi level of any conducting phase a is then synonymous with the electrochemical potential of an electron in that phase, i.e. E = fi . 

The solution phase of an electrochemical cell does not contain free electrons, but it does generally contain a redox couple which can equilibrate with the free electrons in the electrode. This allows us to extend the concept of the Fermi level to the solution. Consider the redox couple 0,R in contact with an electrode el at which the redox reaction 0 H- nt R occurs. When the system is at equilibrium, the work of transferring an electron across the electrode/solution interface to transform (l/n)0 to (l/n)R is zero, i.e. 

Now // i (equilibrium) is the Fermi level of any electrode that is in equihbrium with the 0,R redox couple. This also defines the Fermi level of the redox couple in solution, to which we shall give the symbol Iso.r. 

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Figure 1.7 Dye-sensitised solar cell (a) cell architecture (b) electronic energy levels. The placement of the semiconductor band-edge energy and the solution Fermi levels S/S, S /S and 1713 on the same scale, the vacuum scale of electronic energy, is explained in Appendix lA at the end of this chapter.

Figure 1.8 Cell schematics for a regenerative solar cell based on (a) an n-type photoelectrode (b) ap-type photoelectrode. The top diagrams show the cell reactions under illumination, the middle diagrams the electronic energy levels and band bending, and the bottom diagrams the cell current-voltage (I-U) characteristics with the photoelectrode and counter electrode (CE) currents shown in the same quadrant. The maximum power point is located at the point on the current-voltage curve at which the rectangle of maximum area may be inscribed in this quadrant. The photovoltage V, the electron and hole quasi-Fermi levels E and fip and the solution Fermi level f o.R, the open-circuit potential Ugc of the photoelectrode and the standard redox potential 17 ° of the 0,R redox couple are also shown.

The concept of the solution Fermi level is discussed in Appendix lA at the end of this chapter. 

THE VACUUM SCALE OF ELECTRODE POTENTIAL AND THE CONCEPT OF THE SOLUTION FERMI LEVEL... 

We are now in a position to relate the electronic energy levels of the solution and the electrode on the same scale. It follows from the definition of absolute electrode potential and its value for the SHE, given in eq. 1A.14, that the solution Fermi level qr of a redox couple 0,R is related to its electrode potential Uq r (SHE) on the SHE scale by... 

Figure 4.24 shows how the band bending adjusts when the potential t/ of an n-type semiconductor electrode is altered. This changes the position of the semiconductor Fermi level E with respect to the solution Fermi level at the same time adjusting the band bending across the space-charge layer, but the valence and conduction band-edge... 

This behaviour is observed only for reasonably defect-free semiconductors. Such a solution semiconductor junction behaves like a metal semiconductor Schottky junction, with the electrolyte solution playing the role of the metal. The other extreme of behaviour, a Bardeen junction in which the band bending is fixed and the band edges float with respectto the solution Fermi level, is observed in solution semiconductor junctions with a high density of interfacial states. 

A Le Chatelier concentration effect raises or lowers the solution Fermi level by the relative concentrations of the reduced and oxidized forms. 

Fig. 8-18. Band edge levels of semiconductor electrodes in a solution of pH 1 (n) = n-type (n, p) = n- and p types REDOX = redox reactions and the standard Fermi level. [From Gleria-Memming, 1975.]...

Figure 9-18 illustrates the band edge levels of compound semiconductor electrodes in aqueous solutions and the equivalent Fermi levels of the following oxidative and reductive dissolution reactions ... 

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... 

Figure 3. Schematic of a semiconductor-aqueous electrolyte solution interface, ignoring band bending, Ec and are the band edges of the conduction and valence bands, respectively, Ef(H20/h2> and Ef(02/h20) are the Fermi levels in the solution for the redox reactions indicated. The quasi-Fermi levels with illumination by light of energy hv are designated Ef and pEi respectively, for electrons and holes (13).

Figure 5. n-Type semiconductor—electrolyte solution interface with a surface depletion layer, in the dark and with two intensities of illumination. Symbols as in Figure 3 and 4 with Ec and E the band edges of the conduction and valence bands, respectively, under illumination, and Ef(H2) Ef(Om) abbreviations for Ef(H20/h2) and Ep(02/H20)y respectively. The quasi-Fermi levels Ei> and pEp are at different positions in the surface region than in the bulk as a result of the limited penetration of light into the interior. Fermi levels in solution as in Figures 3 and 4(13). 

The photoinduced difference between the quasi-Fermi level for electrons in the Ti02 and the quasi-Fermi level for holes in solution, EFn = EFn, - EFp.solution, sets an upper limit to the photovoltage, Voc, because it is this potential difference and the fact that electrons and holes are confined to separate chemical phases that drives electrons toward the substrate electrode and holes toward the counterelectrode. Although VEFn is mainly comprised of V x in DSSCs, there is, nevertheless, a possible role for q at interfaces where the field cannot be entirely screened by mobile electrolyte. 

In the liquid environment the situation is different from that described above. Highly doped electrodes are often avoided because they present an enhanced reactivity with the solution. The sample and tip biases are set independently with respect to levels in solution and the position of the band edges of the semiconductor is fixed with respect to the tip Fermi level iU is indeed generally fixed within a narrow potential window). It follows that the situation is often the one shown in Fig. 6, with the tip Fermi level located in the band gap. In this situation, maintaining the tip at a constant height above the surface requires that the n-type electrode be cathodically biased so as to provide a sufficiently large density of electrons (Fig. 6 c). The stability of this situation is governed by the sample bias and not by the tip bias since the position of the tip Fermi level is not critical here, unless it is outside the band gap. 

Similar to for the energy levels in semiconductors, the energy levels of electrons in electrolytes associated with ions are characterized by the redox potential, Eraiox-The redox potential describes the tendency of the species to give up or accept electrons and can be considered as the effective Fermi level of the solution. 

The difference between the flatband potentials ofp-Si and n-Si plus the differences between the bulk Fermi level to the corresponding band edges equal the band gap, 1.12eV when the band edges of the two materials are the same in the solution. Such situations have been observed." " However, in many situations the measurement of flatband potentials ofp-Si and n-Si does not yield the band gap. There are two possible explanations. In one the band edges of p -Si and n-Si may not have the same energy... 

The cell delivers cnrrent / at voltage V. Neglecting the effect of internal resistance, qV is the difference between the quasi-Fermi level ftp of the illuminated photoelectrode and the Fermi level o,r of the redox solution. The counter electrode should be virtually nonpolarised so its potential departs httle from when current is drawn. The two electrodes need not have the same area but must pass the same current. 

Moreover, the use of the term Fermi level for non-fermionic solution species has been criticised by Reiss (1985) but see also Pleskov and Gurevich (1986), pp. 58-59. 

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