We will now examine what happens when a semiconductor is placed in an electrolyte solution. Initially, we will consider the situation at open circuit under dark conditions (i.e., no photoexcitation of electrons). We will then examine how the interface that is established influences the flow of current. Finally, we will look at the impact of light on current flow.

As described earlier in Chapter 3, when a metal is placed in an electrolyte solution containing a redox couple, electron transfer occurs due to a difference in the energy of electrons in the metal and that of electrons associated with electroactive species in the electrolyte. If there is a net transfer of electrons from the metal to solution-containing species, the metal is left with a positive charge that is balanced by a negative charge on the solution side of the interface. The charge on the metal resides at the surface of the electrode. In contrast, the charge on the solution side is distributed throughout the double layer. For solutions with a significant concentration of ions, the thickness of the double layer is quite small, the diffuse portion of the double layer is not important, and nearly all of the charge resides in the Helmholtz layers.

The situation for a semiconductor is somewhat different. For illustration purposes, we consider an n-type semiconductor whose Fermi energy is higher than that of the redox couple in solution, as illustrated in Figure 15.10a, where all energy bands in the semiconductor are shown to be flat prior to contact with the electrolyte. When the n-type semiconductor electrode is brought into contact with the electrolyte under open-circuit conditions, the higher energy electrons in the semiconductor move to the interface where they react with the redox couple. This net transfer of electrons continues until the electron energy is the same in both materials; in other words, the electron transfer continues until the Fermi level of the semiconductor is equal to the reversible potential of the redox couple. Up to now, it may appear that the semiconductor behaves just like a metal. However, the situation is actually quite different. A doped n-type semiconductor consists of immobile, positively charged donor atoms and mobile electrons, as described above. The net transfer of electrons out of the semiconductor results in a depletion zone near the interface where the semiconductor is depleted of electrons. This region is also commonly referred to as the space-charge region, reflecting the fact that migration of the free electrons into the solution has left behind positively charged donor ions  as shown in Figure 15.10b.

The net positive charge in the depletion region is balanced by negative charge on the solution side of the interface (Figure 15.10b). Because the double layer is thin relative to the thickness of the depletion region, the negative charge on the solution side lies very close to the interface. As described by Poisson’s equation, the local imbalance of charge results in an electric field and bending of the semiconductor bands within the depletion region, as shown in Figure 15.10c. The electric field and the associated band bending are greatest at the electrolyte–semiconductor interface, and the potential difference between the energy bands in the bulk and at the interface of the semiconductor is the space-charge voltage (V_SC).

In this chapter, we consider the ideal case where the entire drop in potential at the interface is across the semiconductor. This assumption provides a reasonable approximation of the behavior of the semiconductor–electrolyte interface in general, and a relatively good approximation for some systems. The physical situation it represents is very different than the metal–electrolyte interface that we have considered up to this point. In the metal–electrolyte system, the potential drop across the double layer was critical to our description of the interface and electrochemical reactions at the interface. Here, we assume that the electrolyte is sufficiently concentrated such that the diffuse part of the double layer is not important (see Chapters 3 and 11). We also assume that the potential drop across the Helmholtz portion of the double layer is sufficiently small, relative to the drop across the space-charge layer, that it can be neglected. This is, in general, a good assumption since the capacitance of the electrolyte double layer is typically much larger than that of the semiconductor; therefore, for a given amount of charge, the potential drop across the Helmholtz double layer will be much less than that across the depletion layer of the semiconductor. Finally, we ignore the thickness of the double layer and assume that the charge on the electrolyte side of the interface is located at the interface. This assumption is reasonable since the thickness of the Helmholtz portion of the double layer is ∼0.5 nm compared to depletion layer thicknesses of 10–1000 nm in the semiconductor. Nevertheless, the physical situation described in this chapter is ideal, and additional factors (e.g., surface states in the band gap) will be important for some systems. A more comprehensive treatment can be found in sources such as those listed at the end of the chapter.

With these assumptions, we will now develop a description for the potential field and capacitance in the space-charge layer. The description is provided for an n-type semiconducting electrode, but analogous equations apply for a p-type semiconductor. This picture will allow us to determine V_SC and the thickness of the depletion layer as a function of the potential and the doping level. To do so, we will use Poisson’s equation, which relates the electric field (gradient of the potential) to the charge density:

(15.3)

Here ρ_e is the charge density (charge per volume) and  is the permittivity of the semiconductor, which is equal to the product of its dielectric constant (ε_r) and the permittivity of free space (ε₀). As a first approximation, we ignore any electrons in the depletion region and assume that all of the charge is due to the fixed donor atoms. Thus, the charge density, ρ_e, is a constant throughout the depletion region and is equal to qN_D. We define x = 0 at the interface, and x = W at the edge of the depletion region. Integration of Equation 15.3 yields

(15.4)

where the subscript int is used to designate the interface. Note that potential at the interface is negative relative to the bulk semiconductor (negative potential equates to higher electron energy versus the vacuum level) for the situation considered here (n-type semiconductor). For a p-type semiconductor, the bands will bend in the opposite manner, and the potential at the interface is more positive than it is in the bulk of the semiconductor.

The voltages V and V_int in Equation 15.4 are with respect to an unspecified reference electrode in solution. To avoid ambiguity, we incorporate the standard definition of overpotential into the equation:

Subtracting the equilibrium value U from the two potentials, Equation 15.4 becomes

As usual,  at the equilibrium potential. Significantly, when the overpotential is zero, there is still a potential difference between the reference electrode and the interface. This potential difference is V_SC,eq, and is an important characteristic of the semiconductor–electrolyte system. It is positive for an n-type semiconductor since the bulk potential is higher than that at the interface at equilibrium (Figure 15.11). To explore this potential further, consider what would happen if we start with the electrode at open circuit and then lower its potential. As the potential of the electrode is made more negative (raising electron energy), the charge in the depletion region and its thickness will get smaller, and band bending will be reduced. We can continue reducing the potential until there is no charge (Q = 0) in the depletion layer and no band bending at all—equivalent to the point of zero charge for metal electrodes. The potential difference across the space-charge layer is zero, and W is zero, but the overpotential is not zero. In fact, the overpotential for the condition where there is no band bending is simply −V_SC,eq. This quantity is called the flat-band potential, V_fb.

(15.5)

We will use the flat-band potential as our reference point, since that is what is typically done. The width of the depletion layer can be expressed as

(15.6)

From Equation 15.6, we see that the width of the depletion layer, W, is a function of both the applied voltage (expressed as the overpotential) and the dopant concentration, N_D. It is valid for .

Under the above assumptions, the charge in the depletion region of the semiconductor is

(15.7)

Equation 15.7 can be differentiated to yield the capacitance per area for an n-type semiconductor:

(15.8)

This equation is a simplified version of the classical Mott−Schottky equation. A linear plot of 1/C² versus  can be used to determine V_fb from the intercept as shown in Illustration 15.3. Additionally, the slope may be used to estimate the dopant density. Note that C² is always positive. Therefore, since the charge density (qN_D) is positive for donor atoms, the slope for an n-type semiconductor is positive, and the equation is valid for .

ILLUSTRATION 15.3

The capacitance of the depletion region of an n-type semiconductor was measured (F·cm⁻²), and the data are plotted in the figure provided. The dopant concentration is 3 × 10¹⁵ cm⁻³, and the dielectric constant of the semiconductor (ε_r) is 11.9. Please determine the voltage drop across and the width of the depletion region at open-circuit conditions.

SOLUTION:

The potential of the semiconductor electrode at equilibrium will depend on the redox couple in solution, and V_fb cannot be extracted from a simple measurement of the potential of the semiconductor electrode relative to a reference electrode in solution. In order to estimate V_fb, we can measure the capacitance as a function of the overpotential. According to Equation 15.8, a plot of 1/C² versus η will yield a straight line. This is called a Mott–Schottky plot. The y-intercept  can be used to find V_fb. Using the data provided for the capacitance, a fit of 1/C² versus η yields a line with a slope of 5.05 × 10¹⁵ and a y-intercept of 2.02 × 10¹⁵.

The y-intercept is equal to −mV_fb, where m is the slope of the line =  (see Equation 15.8. Therefore, .

We can now use  to estimate W, the width of the depletion region. According to Equation 15.6, and noting that ε = ε_rε_o,

For an electrolyte with ion concentrations of 0.1 M, the electrochemical double layer is on the order of 1 nm thick, which is more than two orders of magnitude smaller than the depletion layer thickness given above.

A similar discussion and derivation can be made for p-type semiconductors, where the band bending is in the opposite direction as shown in Figure 15.12. The Fermi level is just above the valence band. The charge density is negative for the acceptor atoms in the depletion region, and V_fb is positive. In addition, the slope of 1/C² versus  line is negative for a p-type semiconductor, and Equations 15.7 and 15.8 apply for .