Chapter 7

Theoretical Model for the Electrochemistry of Boron Doped Diamond

 

7.0       Outline

 

·            This chapter develops a theoretical model for the electrochemistry of boron doped diamond.

 

·            The relationship between current density and applied potential is investigated

 

·            The AC impedance results are considered in terms of the surface state model.

 

 

7.1       Interpreting the Metallic Behaviour of Highly Doped Diamond

 

Chapter 5 showed that highly doped diamond films, with boron concentrations over 1020 cm-3, exhibited metallic behaviour. Cyclic voltammograms of well known redox couples showed both oxidation and reduction peaks.

 

Two possible explanations for this behaviour are that:

 

·          The samples are degenerately doped (see section 4.3).

 

·          Contamination of the specimens provided a conduction path through the films. Feasible sources of contamination include tantalum (Ta) from the filaments in the CVD reactor and graphite (C) formed during diamond deposition.

 

These explanations can be discounted because:

 

·          A Mott-Schottky regime was observed, as shown in section 5.7 (figure 5.17).

 

·          Two time constants were observed in impedance measurements of the films, as shown in sections 5.8 – 5.9 (figures 5.24 – 5.25).

 

This implies that the electrode was a p-type semiconductor and the electron transfer occurred via a surface state.

 

7.2       Explanation of the Metallic Behaviour of Highly Doped Diamond

 

To explain the behaviour of boron doped diamond, the positions of the conduction band maximum, the valence band minimum and the Fermi level relative to the energy levels of the redox species in solution need to be considered.

Figure 7.1 – Schematic energy level diagram showing the band edges of the semiconductor and the energy levels of redox species. Redox potentials from reference 12.

 

Figure 7.1 gives a schematic representation of the relevant energy levels. The band edges have been calculated from the work function of the semiconductor, assuming that there was no voltage drop across the space charge layer. 119(b)

 

7.2.1    Hydrogen Terminated Diamond

 

For hydrogen terminated diamond, the valence band maximum lies at a higher energy than the proton redox couple. Therefore, in aqueous solution, holes will accumulate at the surface.

 

A surface charge on the semiconductor and potential drop across the Helmholtz layer will result in the material behaving in a metallic manner.

 

Figure 7.2 provides a schematic diagram of the contact process using the standard hydrogen reaction as an example.

 

Figure 7.2 - Reaction at a Hydrogen Terminated Surface

 

7.2.2    Oxygen Terminated Diamond

 

The diamond samples, being polycrystalline in nature, contained a high density of grain boundaries. Grain boundaries are known to provide trapping states. 119(c) The grain boundaries in CVD diamond can be considered to be graphitic in nature. Therefore, the energy of the grain boundary states will be approximately –5.0 eV (the energy level of graphite). This energy level is comparable to the energy levels of the redox couples in aqueous solution.

 

Figure 7.3 provides a schematic diagram of the contact process for an oxygen terminated surface which involves the surface states. The diagram shows the case where the potential drop across the Helmholtz layer is constant. When this is not the case, any change in the potential across the Helmholtz layer would lower the level of the redox couple in solution relative to the valence band maximum.

 

 

Figure 7.3 - Reaction at an Oxygen Terminated Surface (SS = surface state)

 

7.3       The relationship between current density and applied potential

 

The model for surface state mediated electron transfer presented in Chapter 4 assumed that the potential drop over the Helmholtz layer was insignificant (§ 4.4.1).  If this assumption is not valid, then the system may be modelled as shown in figure 7.4.

 

The Helmholtz region and the space charge region can be considered to be connected in series. Each region acts as a resistor in parallel with a capacitor.

Figure 7.4 – Equivalent circuit for surface mediated transfer with significant potential drop across the Helmholtz region

 

Applying Kirchhoff’s first law:

 

                                                            (equation 7.1)

 

                                                                                (equation 7.2)

 

Using equation 7.2 to eliminate dVsc from equation 7.1:

 

     (equation 7.3)

                                                                            Qss = charge on the surface states.

 

Let r(E0) be the density of states per unit energy (eV):

 

                                                                                  (equation 7.4)

Hence:

                                           (equation 7.5)

 

or                                                                (equation 7.6)

                                                             where .           (equation 7.7)

 

Assuming that the capacitances, CH and CSC, and the density of states, r(E0), are independent of the potential:

 

                                                                        (equation 7.8)

 

and                                                                         (equation 7.9)

 

Substituting equation 7.9 into equation 4.8 (the Butler-Volmer equation for
a = ½):

 

                                                                                              

 

                                                   (equation 7.10)

 

                                    (equation 7.11)

 

The current across the space charge region, jSC can be modelled as a Schottky diode (substitute equation 7.9) into equation 4.18:

 

                                 (equation 7.12)

 

Combining equations 7.11 and 5.12 above, gives an expression for the total current, j, that varies with the overpotential, dV, and the currents across the space charge and Helmholtz regions, jSC and jH.

 

              (equation 7.13)

 

At steady state, the stored charge will be constant and the current through the Helmholtz layer must equal that through the space charge region.

 

                                            j = jH = jSC                                                 (equation 7.14)

 

Therefore, equation 7.13 can be rewritten:

             (equation 7.15)

 

Equation 7.15 is an expression for the current density, j, that varies with the applied overpotential, dV. All the other terms are constant. An analytical solution to the equation can not be obtained and so numerical methods are necessary.

 

Mathcad 8 Professional (MathSoft, Inc.) was used to find numerical solutions. The software automatically selected an appropriate algorithm to solve a given equation. Non-linear solutions are found using the Levenberg-Marquardt, Quasi-Newton and Conjugate Gradient algorithms. The solutions found by numerical means were not necessarily unique. 139-142

 

As stated in section 4.4.2, it is usual in semiconductor electrochemistry to assume that the capacitance of the Helmholtz layer to be much greater than that of the space charge region (CH >> CSC). If this is the case, then g tends to unity (g ® 1) and the equation can be solved analytically.

 

When g = 1, then equation 7.15 simplifies to equation 7.16.

 

                                                                                                             (equation 7.16)

7.3.1    Special cases of the current density - applied potential relationship

 

7.3.1.1        The case where |j0H| >> |j|

 

For the case where |j0H| >> |j|, then several terms in equation 7.16 become negligible. This leads to the much simpler relationship given in equation 7.17.

 

                                                                       (equation 7.17)

 

Figure 7.5 shows a Mathcad plot of equation 7.16 for the case where |j0H| >> |j|. The behaviour is independent of b and is typical of a p-type semiconductor.

 

Figure 7.5 – Equation 7.16 plotted for the current density range –1 ´ 10-3 A cm-2 to 1 ´ 10-3 A cm-2 Case where |j0H| >> |j|

 

7.3.1.2        The case where |j| >> |j0H|

 

For the case where |j| >> |j0H|,

 

                                                                                                             (equation 7.18)

 

For positive j:

 

                                 (equation 7.19)

 

For negative j:

 

                                                                                         (equation 7.20)

 

Therefore, at negative potentials, the current density is independent of the applied potential.

 

Figures 7.6 and 7.7 show a Mathcad plots of equation 7.16 for the case where |j0H| << |j|. Figure 7.6 shows the case for b = 1. Figure 7.7 shows the case for b = 1000.

 

Figure 7.6– Equation 7.16 plotted for the current range -1 ´ 10-3 A cm-2 to 1 ´ 10-3 A cm-2 Case where |j0H| << |j|,    b = 1

 

Figure 7.7– Equation 7.16 plotted for the current range  -1 ´ 10-3 A cm-2 to 1 ´ 10-3 A cm-2 Case where |j0H| << |j|,    b = 1000

 

7.3.1.3        The case where |j0SC| >> |j| and j » j0H

 

For the case where |j0SC| >> |j| and j » j0H, equation 7.16 simplifies to equation 7.21.

 

                                                     (equation 7.21)

 

Figure 7.8 shows a Mathcad plot of equation 7.16 for the case where |j0SC>|j|. The behaviour is independent of b and is classic metal behaviour.

 

Figure 7.8– Equation 7.16 plotted for the current range -1 ´ 10-3 A cm-2 to 1 ´ 10-3 A cm-2 Case where |j0SC>|j|

 

7.3.2    The current density - applied potential relationship at intermediate values

 

At high doping levels, j0SC will be high. Metal behaviour will therefore be observed.

 

Reducing the doping level, implies that j0SC will become more significant and semiconductor behaviour would be expected.

 

Figures 7.9 and 7.10 show Mathcad plots of equation 7.16 over current density ranges of -1 ´ 10-4 A cm-2 to 1 ´ 10-4 A cm-2 and –1 ´ 10-3 A cm-2 to 1 ´ 10‑3 A cm-2 respectively.  For both plots values of j0H = 1 ´ 10-5 A cm-2 and j0SC = 1 ´ 10-1 A cm-2 were used.

 

Figures 7.11 and 7.12 show Mathcad plots of equation 7.16 with various different values for the constants j0H, j0SC and b.

 

Figure 7.9 – Equation 7.16 plotted for the current range -1 ´ 10-4 to 1 ´ 10-4 A Number of surface states = 1013,   Helmholtz capacitance = 1 mF

 

Figure 7.10 – Equation 7.16 plotted for the current range -1 ´ 10-3 to 1 ´ 10-3 A Number of surface states = 1013,   Helmholtz capacitance = 1 mF

 

Equations and constants used to plot Figures 7.11 and 7.12

 

Figures 7.11  & 7.12 – Mathcad plots of equation 7.16.

 

7.4       AC Impedance and the Surface State Model

 

The surface state model can be extended to consider small amplitude modulations (symbolised by overtilde embellishments):

 

                                                                                        (equation 7.22)

 

Applying Kirchhoff’s law:

 

                                                  (equation 7.23)

 

                                                                            (equation 7.24)

 

The general equation for the current density across Helmholtz current, jH, is:

(see § 4.4.4, equation 4.8)

 

                                                                      (equation 7.25)

 

Applying a small modulation gives:

 

                                                   (equation 7.26)

 

Simplifying using the following identities:

,         exp(x) ≈ 1 + x        and      cosh2 x = sinh2 (x + 1)

 

gives                                                                (equation 7.27)

 

For a steady DC current, j = jH

 

                                      (equation 7.28)


A general equation for the current density across the space charge layer can be derived from equations 4.18 and 7.9.

 

                                                                                                             (equation 7.29)

 

Expanding for small perturbations:

 

                   exp(x) ≈ 1 + x,            exp(y) ≈ 1 + y                  xy → 0

 

                                                                                                             (equation 7.30)

 

For a steady DC current, j = jSC

                                                                                                             (equation 7.31)

 

                                                              (equation 7.32)

 

       where                and          

 

                                                                                             (equations 7.33 and 7.34)

 

For a particular potential, j, b, j0,H and dVSC are all constant.

 

dV, dVSC and j can be determined from steady-state experiments.

 

Equations 7.22 and 7.32 can be combined in order to lead to an expression which eliminates and .

 

Equation 7.22:                             

 

                                                 where         

                                                                                                (equations 7.35 & 3.36)

 

                                                                               (equation 7.37)

 

                                                                            (equation 7.38)

 

Substituting equation 7.38 into equation 7.22 gives:

 

                                                  (equation 7.39)

 

Substituting equation 7.37 into equation 3.65 gives:

 

                                                                            (equation 7.40)

 

                                                                          (equation 7.41)

 

Substituting equation 7.22 into equation 7.39 gives:

 

                                                                                                             (equation 7.42)

 

                                                                                                             (equation 7.43)

 

Substitute for  ((equation 7.41) in the equation 7.43:

 

                                                                                                             (equation 7.44)

 

                                        (equation 7.45)

 

 

Which leads to an expression for the impedance:

 

                                                                                                  (equation 7.46)

 

                                            (equation 7.47)

In Summary:

 

 

                                                  (equation 7.47)

 

 

where

 

 

                                                                                                            (equation 7.33)

 

 

                                                  (equation 7.34)

 

 

                                                                                   (equation 7.36)

 

 

                                                                                                     (equation 7.3)

 

 

                                                                                                    (equation 7.7)

 

 

If  then g will tend to unity (γ → 1).

 

Considering the case where j = 0, the expression in equation 7.47 simplifies to:

 

                                                            (equation 7.48)

 

                                                                (equation 7.49)

 

This expression for Z forms two semicircles when plotted in the complex plane. The first semicircle has a radius of 1/B with a maximum, wmax, of B/CSC. The second semicircle has a radius of 1/D and a maximum, wmax, of D/CH.

 

When not at open circuit potential, the behaviour is slightly more complicated and it is necessary to find values of j, dV and dVSC from DC experiments.

 

In Summary:

 

 

                                                                 (equation 7.49)

 

 

where

 

 

                                                    (equation 7.34)

 

 

                                                                                                         (equation 7.50)

 

 

                                                                                                              (equation 7.51)

 

 

Figure 7.13 shows a current density – potential plot for a given set of parameters.  That the curve passes through the origin and the currents are significant for both small positive potentials and small negative potentials. This would correspond to a system where the current density was transport controlled and reversible cyclic voltammetry would be observed.

 

Figure 7.14 shows an impedance calculation at open circuit. Two semicircles can be seen.  Figure 7.15 shows various impedance curves taken for various jdV values taken from the plot in figure 7.13.

 

Figure 7.13  –  current density / potential plot

 

Figure 7.14  –  impedance plot for open circuit conditions

 

Figure 7.15  –  impedance plots at various potentials

 

7.5       Interpreting the Semiconductor Behaviour of Low Doped Diamond

 

The chapter 6 showed that low doped diamond films, with boron concentrations of the order of 1019 cm-3, exhibited semiconductor behaviour.

 

For oxygen terminated diamond films, cyclic voltammograms of well known redox couples were irreversible with no defined reduction peaks. The diamond electrode acts as a p‑type semiconductor. At negative applied potentials, the p‑type semiconductor is reversed biased and the current density will be limited.

 

For hydrogen terminated diamond films, the valence band is at a higher energy. The valence band minimum at the surface is higher than the Fermi level (see figure 7.1).  Accumulation of holes at the surface may occur and so both anodic and cathodic reactions are possible.

 

7.6       Summary

 

The standard theories for metal or semiconductor electrochemistry, as outlined in chapter 4, are not sufficient to provide a model for the electrochemistry of boron doped polycrystalline diamond films.

 

The mechanism of charge transfer must be considered. The presence of surface states explains the difference in behaviour between hydrogen and oxygen terminated diamond films.

 

A relationship between the current density and applied potential has been derived. This agrees with the experimental results, which show that metallic behaviour predominates in highly doped diamond samples, while semiconductor behaviour is seen in low doped samples.

 

The surface state model was applied to the AC impedance studies and an expression was obtained that predicts the two time constants that have been seen in the experimental results.