As it was already mentioned, the Fermi energy is a very important parameter for solids as well as for electrolyte solutions. Namely, the Fermi level is an 'indicator' showing how the energy levels are occupied by electrons.
From thermodynamics it is known that two phases are in equilibrium if their electrochemical potentials are equal. The electrochemical potential of a redox couple in solution is determined by Eredox, while the electrochemical potential of the electrons in the semiconductor is determined by the Fermi level EF. If Eredox and EF are different, a charge redistribution between semiconductor and solution is required in order to equilibrate the two phases. In order to find how the Fermi level of the electrode depends on the ion redistribution in solution let define in a more rigorous way the chemical and electrochemical potentials.
For an ideal solution, the chemical potential of an ion i is described by:
where ci is the molarity of the solute.
For a real solution where ion-ion interactions are considered, the Equation 7 does not describe the chemical potential adequately, therefore a correction parameter to the molarity is usually defined, and Equation 7 is transformed into
where γi is the correction parameter to the molarity, ai=γiŚci is called the activity of the ion i.
The electrochemical potential of species is defined by means of the chemical potential as follows:
where F=96489 C/mole is the Faraday constant, Zi is the charge of the ion and φ is the potential in volts. On the other hand, the general thermodynamical definition of the electrochemical potential is defined as follows:
where G is the Gibbs free energy and ni is the concentration of the i species. Equation 10 assumes that the temperature, pressure and concentrations of other species than i are kept constant.
As it was already mentioned above, the electrochemical potential is constant at equilibrium in all contacting phases:
In addition, for a reaction in equilibrium like the following one
where B's are the species in reaction and a's are integers, the rate of reaction in both directions must be equal. It can be shown that at equilibrium the electrochemical potentials of the reactants and products are related by:
Now let consider a reaction where an electron is exchanged between an electrode and a redox couple B1/B2.
According to Eq, 13, one can write:
where μe is the electrochemical potential of the electrons in the electrode, which is nothing else but the Fermi energy of the electrode.
Thus, by using Eq. 15 we get the Fermi level in the solid to be equal to
and φsol is the potential in solution. Note that in order to obtain Eq. 16, Eqs. 8 and 9 were substituted in Eq. 15.
Taking into account that the electrochemical potential is measured in joule/mole and the Fermi energy in volts/electron, one can write
In the Eq. 19 the activities have been substituted by concentrations of the corresponding species. E'B is a property of the redox couple B1/B2 and reflects the tendency to inject or extract electrons from the electrode.
Thus, the Fermi level in the electrode is determined:
From Equation 19 it is easy to observe that E'B is difficult to estimate, because moi is not so easy to determine (see Eq. 18). What is relatively easy to measure is the difference between the Fermi levels of two electrodes, one in contact with a redox couple A>1/A2 and the second electrode in contact with a redox couple B1/B2, where φsol is constant through the solution. According to Eq. 19, the measured voltage difference between these two electrodes will be
In order to give Eq. 20 a practical use, one redox couple is taken to be standard and is defined as a reference redox couple. Thus, Vm of other redox couples is measured relative to this reference redox couple. Namely, the hydrogen couple (H+/H2 ) is usually chosen as reference redox couple, and its corresponding Eredox energy level is chosen as the arbitrary zero.
Therefore, if a redox couple has a stronger tendency to inject electrons into an electrode than the hydrogen couple, this redox couple will have a negative redox potential. And vice versa, if the redox couple has a weaker tendency than hydrogen couple to inject electrons into an electrode, then the redox potential of the redox couple will be considered positive.
Thus, if in Eq. 18 the couple B1/B2 will be considered to be H+/H2 and the concentrations of hydrogen species are equal [H+]=[H2] or, in other words, the activities of the hydrogen species are considered to be unity, then Eq. 20 becomes
Now if one considers the Fermi energy of the electrode in contact with H+/H2 to be our zero reference and consequently we set Eref=EF2=0, then one obtains
Equation 22 gives the value of the Fermi energy of an electrode which is in direct contact to a redox couple A1/A2, relative to the hydrogen reference electrode. In the Eq. 22 we introduced the parameter Eo=E'A-E'ref which is called 'the standard redox potential' of the redox couple A1/A2 and is tabulated in handbooks. As a definition for the standard redox potential of a redox couple A1/A2 the following formulation can be taken:
The experimental setup that is mainly used to determine the standard redox potential of a redox couple (see Figure 2.9 on page 25) consists of an electrochemical cell with two compartments. In one compartment are H+/H2 species in contact with a metal, this is the reference redox couple at standard conditions. The second compartment contains the redox couple for which the standard redox potential shall be measured, for example Cl2/Cl- at unit activities, also in contact with a metal. In order to have a constant potential throughout the solution a salt bridge is used, which is blocking the passage of active species, but passes a current flow of inert ions, which has negligible effect on the potential. Having these requirements satisfied the voltmeter is measuring the voltage with respect to the reference redox couple which can be considered as the "ground". The voltage measured by the voltmeter is the standard electrode potential of the Cl2/Cl- redox couple.
Figure 2.9: A schematic illustration of the principle for measuring the standard redox potential. In our case the redox couple is Cl2/Cl-. The installation contains a double room cell, where in one of the rooms a metal in contact with the solution of interest at unit activity is present, whereas in the second one we have a metal in contact with a hydrogen containing solution at standard conditions. The two rooms communicate electrically by means of a salt bridge. The voltmeter is measuring the difference in potential on these two metals. The measured value is actually the standard electrode potential of the redox couple. Thus, the zero of energy is the Fermi level of the metal in contact with hydrogen under standard conditions.
Thus, the standard redox potential of a redox couple is showing the tendency of this couple to inject or to extract electrons from a solid. If the couple is tending to inject electrons into the solid then the standard redox potential will be negative, if the couple is tending to extract electrons then it will be positive.
It should be noted that the physicists, when dealing with solid/gas interfaces, usually take as an energy reference the energy of an electron at infinity (Ee). The chemists or better saying electro-chemists take as an energy reference the Fermi energy of the hydrogen electrode under standard conditions as described above. In order to have a clear picture of the solids immersed in solutions it is required to have a relation between these two reference energies. Several derivations of the relation between the two zeros of the energy have been proposed. The results show that the hydrogen reference electrode is below Ee by around 4.5 eV [20, 21, 22]. Taking into account that the energies on the hydrogen scale are measurable quantities and are available in handbooks, it is advisable to use this scale.