Chapter 4

Standard Electrochemical Theory


4.0       Outline


·          This chapter outlines the standard electrochemical theory on which this study was based.


·          A comparison is made between the behaviour of metal and semiconductor electrodes.


·          An expression is derived relating the steady state current to the overpotential for surface state mediated charge transfer.


4.1       Metal Electrochemistry


In this section the Marcus-Gerischer theory for electron transfer at a metal electrode is outlined. Reference 119(a) contains details of the theory and lists the original references and review articles.  Underlying this approach is the idea that electron transfer occurs between occupied and unoccupied states that are equal in energy. 120


The total current density, j, at a metal-solution interface is the sum of the anodic current density, ja, and the cathodic density, jc. By convention, the cathodic current is negative.


                                                          j = ja + jc .                                     (equation 4.1)


Anodic current flows when oxidation occurs at the electrode surface:


                                      R → O + e-                                 (e.g. Fe2+(aq) → Fe3+(aq) + e-)


Cathodic current flows when reduction occurs at the electrode surface:


                                       O + e-→ R                                  (e.g. Fe3+(aq) + e- → Fe2+(aq))


In the Marcus-Gerischer approach to electron transfer it is assumed that the Born-Oppenheimer approximation applies, i.e., the nuclei, being so much heavier than the electron, move relatively slowly and may be treated as stationary during electron transfer. 121 Hence electrons are transferred to energy levels at or near the Fermi level in the metal.  At a metal electrode a change in the applied potential results in charging of the metal which leads to a change in the potential at the surface.  As the energy of the surface electrons is altered the position of the Fermi level changes.  However, the occupation of states about the Fermi level remains constant.  As a result the current densities are simply proportional to the overlap integral of the Fermi function in the metal and the density of states function for the solution species:


ja µ (overlap of EF and ER)


jc µ - (overlap of EF and EO)


Figure 4.2 shows a schematic energy diagram for a metal-solution interface.


At the equilibrium potential, the overlap integrals are equal. The values of ja and jc are therefore equal and opposite and the net current density is zero. This situation is shown to the left in figure 4.2.


At a positive overpotential, the overlap integral for the reduced species is greater than that for the oxidised species. The net current density is therefore positive. An anodic current flows and oxidation of the reduced species occurs. This situation is shown in the upper right diagram in figure 4.2.


At a negative overpotential, the overlap integral for the oxidised species is greater than that for the reduced species. The net current density is therefore negative. A cathodic current flows and reduction of the oxidised species occurs. This situation is shown in the lower right diagram in figure 4.2.


Figure 4.1 shows a sketch of current density – overpotential relationship for the case where the transfer coefficient, a = ˝.

Figure 4.1 – current density as a function of overpotential for a metal



Figure 4.2 – Schematic Energy Diagram for a Metal-Solution Interface

4.2       Normal p-type Semiconductor Electrochemistry


As for a metal electrode, the total current density, j, is the sum of the anodic and cathodic current densities (ja & jc). The majority charge carriers in a p-type semiconductor are positive holes in the valence band.


The anodic reaction:


                                   R + h+(VB) → O                         (e.g. Fe2+(aq) + h+(VB) → Fe3+(aq))


The cathodic reaction:


                                   O → R + h+(VB)                         (e.g. Fe3+(aq) → Fe2+(aq) + h+(VB))



The Born-Oppenheimer approximation still applies but unlike the case for a metal electrode, charge transfer can not occur at the Fermi level if the density of states is zero at that energy (i.e. it is in the band gap of the semiconductor). Charge transfer will occur at the surface energy of the valence band, EVB,S.


Current flow at a semiconductor electrode is dependent of the surface concentration of majority charge carriers.  For a p-type semiconductor, the concentration of holes at surface is a significant factor in the rate of charge flow in the anodic reaction:


                               ja µ   (overlap of EVB,S and ER) ´ ( [h+]surface )


Whilst for the cathodic reaction the availability of electrons in the valence band is essentially constant:


                               jc µ - (overlap of EVB,S and EO)



Figure 4.3 – Schematic Energy Diagram for a Semiconductor-Solution Interface

The overlap terms at the valence band edge are not affected by changes in potential. The cathodic current density will therefore remain constant.


jc = jc,0


At a positive overpotential, there will be a higher surface concentration of holes than at equilibrium. There is an exponential relationship between the concentration of holes and the overpotential. Therefore, at positive overpotential, the anodic current density will be greater than at equilibrium.


ja > ja,0


At negative overpotential, the concentration of holes at the surface will be reduced and as the magnitude of the negative overpotential is increased, the concentration will tend to zero. The anodic current density, ja, will therefore fall from its equilibrium value, ja,0.


The total current density, j, is the sum of the constant cathodic current density, jc,  and the variable anodic current density, ja. At increasing positive overpotential, the total current density will rise rapidly, while at decreasing negative overpotential, the total current density will fall to a constant minimum value, equal to the equilibrium cathodic current density, jc,0.

Figure 4.4 – current density as a function of overpotential for a semiconductor


At sufficiently negative potentials, inversion can occur. This phenomenon occurs when band bending at the surface of the semiconductor is sufficiently great that the Fermi level lies in the conduction band. There will then be a finite probability of finding a conduction band electron at the surface of the semiconductor. An anodic current can flow and the total current density will be reduced in magnitude ( jc,0 < j < 0 ).


Anodic current flow due to inversion:


                                    O + e-(CB)→ R                                (e.g. Fe3+(aq) + e-→ Fe2+(aq))



4.3       Highly doped semiconductors


At high doping levels, the semiconductor becomes degenerately doped. The Fermi Level of the semiconductor lies within the valence band (VB) and the space charge region becomes very small. Electrons are able to tunnel from the bulk VB directly into the electrolyte. Therefore, the concentration of available charge carriers is constant and the electrode behaves like a metal.


ja µ (overlap of EF and ER)


jc µ - (overlap of EF and EO)


Cyclic voltammograms will therefore show both oxidation and reduction peaks.


Figure 4.5 shows a schematic energy diagram for a highly doped semiconductor-solution interface.


Figure 4.6 shows a sketch of current density – overpotential relationship.


Figure 4.5 – Schematic Energy Diagram for a Heavily Doped Semiconductor-Solution Interface

Figure 4.6 – current density as a function of overpotential for a heavily doped semiconductor

4.4       Surface State Mediated Electrode Transfer


Electron transfer via surface states, see figure 4.7, is often invoked to explain anomalies in semiconductor electrochemistry. 122(a)

Figure 4.7 – electron transfer via a surface state


In the literature the development of a theory for this mechanism of charge transfer has been almost totally neglected.  The exceptions are the works of Chazalviel and Vanmaekelbergh 123-124 where surface state mediated charge transfer at n-type semiconductors are considered.   In this section the equations that govern the steady-state current at a p-type semiconductor electrode at which electron transfer occurs via surface states are developed.  The theory is based on Marcus-Gerischer model.  To facilitate the development of the theory a number of assumptions were made these are detailed below.


4.4.1    Contributions to the Applied Potential


The change in applied potential across the electrode-solution interface, dV, has two contributions: the potential difference across the space charge region in the solid, dVSC, and the potential difference across the Helmholtz layer in the solution, dVH.


                                                     dV = dVSC + dVH                               (equation 4.2)


As shown in chapter 3, care was required to ensure that electrical contacts to the electrode provided an Ohmic response and did not contribute to the measured potential.



4.4.2    Contributions of the Helmholtz Layer and the Space Charge Region


The capacitance of the Helmholtz layer, CH, is significantly greater than that of the space charge region, CSC. This implies that changes to the potential drop across the Helmholtz layer, dVH, are only due to changes in the surface charge, dQ.




                                                                                                               (equation 4.3)



4.4.3    Electrical Charge at the Surface of a Semiconductor Electrode


The change in the surface charge is dependent on the change in the Fermi level at the surface, E0, and the density of states at the surface, rss.


                                                                                                               (equation 4.4)



                                                                                                               (equation 4.5)



If the distribution of surface states is uniform, then the equation becomes:


                                                                                                               (equation 4.6)


4.4.4    The Butler-Volmer Equation


Electron transfer across the Helmholtz layer is governed by the Butler-Volmer equation. 120


Under non-equilibrium conditions, there will be a difference in energy between electrons in the surface states and the redox couple. The Butler-Volmer equation relates the current density to the overpotential, h.


Assuming efficient mass transport:


                                            (equation 4.7)


If a = ˝,

                                          (equation 4.8)


4.4.5    Schottky Diode


The bulk semiconductor - surface state junction may be modelled as a Schottky diode. 125


                                                                                                               (equation 4.9)


                        where   jSD   =   forward biased current

                                    VD   =   the potential drop between the space charge and the bulk semiconductor (the forward bias)

                                    jB   =   barrier height

                                    A*   =   the “effective Richardson constant” (units of A K‑2 cm‑2)


Figure 4.8 shows the relationship between forward bias and conventional electrochemical current for n-type and p-type semiconductors.

In an n-type semiconductor, forward bias implies a negative applied potential.


Forward bias current is reducing and is therefore, in electrochemical terms, negative.



                            (equation 4.10)


In a p-type semiconductor, such as boron doped diamond, forward bias implies a positive applied potential.


Forward bias current is oxidising and is therefore, in electrochemical terms, positive.


                            (equation 4.11)

Figure 4.8 – Forward Bias and Conventional Electrochemical Current

VD is the potential drop between the semiconductor surface (SS) and the semiconductor bulk (SC).

4.4.6 Model


The figure 4.9 below depicts the influence of applying a positive potential to a semiconductor interface at which there is a high density of surface states.


Figure 4.9 – Schematic Energy Diagram at positive potential


From the diagram expressions relating the potential drop across each component are obtained.


                                                                                                             (equation 4.12)


                                                                                                             (equation 4.13)


                                                                                                             (equation 4.14)


Substituting the expression for the overpotential (equation 4.12) into the Butler-Volmer equation  (equation 4.6) yields:


                                                                                                             (equation 4.15)



Whilst treating the semiconductor / surface state junction as a Schottky-diode results in the expression:




                                                                                                             (equation 4.16)




                                                                                                             (equation 4.17)


                                                                                                             (equation 4.18)



4.5       Steady State Current


As at steady-state                              jSC = jH = j ,                                 (equation 4.19)


an expression can be derived for the steady state current.  Substituting equation 4.6 in to equation 4.15 gives:


                                                            (equation 4.20)


                                                              (equation 4.21)



                                  (equation 4.22)



                               (equation 4.23)



                             (equation 4.24)



                           (equation 4.25)



                                                                                                             (equation 4.26)










                                                                                                             (equation 4.27)








                                                                                                             (equation 4.28)

4.6       Summary


·          Metal electrodes exhibit a symmetrical response to applied potential. Oxidation and reduction peaks can both be seen in cyclic votammograms taken with metal electrodes.


·          p-type semiconductor electrodes are expected to show oxidative peaks, as there will be a forward bias at positive overpotential. The rate of reduction at negative overpotential should reach a limiting value where the total current density is equal to the equilibrium cathodic current density.


·          High doping levels can lead to degenerate doping conditions where the semiconductor will exhibit metallic behaviour.


·          An expression may be derived to relate the overpotential to the steady state current for surface-state mediated charge transfer at p-type semiconductors.