The space-charge layer introduced in the previous section plays a critical role in the flow of current for a semiconductor electrode. The band diagrams that correspond to open-circuit, positive overpotential, and negative overpotential for n-type semiconductors are shown in Figure 15.13. At open circuit , the net flow of current is zero and the conduction band energy increases at the interface as reflected in the band bending illustrated in Figure 15.13a. For positive overpotentials, , the band bending is increased, and the width of the depletion region increases (Figure 15.13b). The band bending has a significant impact on current flow as will be discussed shortly. For negative overpotentials, Figure 15.13c, the band bending decreases or is reversed for n-type semiconductors.

For the ideal semiconductors that we are considering, the band edge positions at the interface are pinned, which means that the edge positions do not change with respect to the solution as the applied potential is varied. In contrast, the relative position of the Fermi level does change, as do the densities of electrons and holes at the surface of the semiconductor. As we’ll see shortly, the density of electrons at the surface plays a key role in understanding the kinetic behavior of semiconductor electrodes. For an n-type semiconductor, the majority charge carriers are electrons in the conduction band, as shown in Figure 15.4, and the concentration of charge carriers in the bulk is n_b, which is approximately equal to N_D. However, the local concentration of electrons is affected by the formation of the depletion region and the associated potential field. Our earlier analysis assumed that all of the electrons were absent in the depletion region, leaving behind only positively charged dopant atoms at fixed sites. We now need to refine our physical model. Although the negative charges on the solution side of the interface make it energetically more difficult to have electrons on the surface of the semiconductor, thermal excitation must be considered. This balance between electrostatic repulsion and random thermal motion is expressed with a Boltzmann factor:

(15.9)

V is the potential of the bulk semiconductor relative to a reference electrode in solution, and V(x) is the potential in the depletion zone at a position x, relative to the same reference electrode. As we saw above for an n-type semiconductor, V(x) is negative relative to the bulk (V). Therefore, the concentration of electrons decreases as you move from the bulk to the semiconductor–electrolyte interface. Altering the potential of the electrode relative to our reference causes the potential difference between the bulk and interface to vary, and thus the concentration of electrons at the interface can change. The surface concentration of electrons can be expressed as

(15.10)

The potential of the interface relative to the bulk reduces to −V_fb when there is no applied potential . We are now ready to examine the electrode kinetics. The heterogeneous electron-transfer reaction at the semiconductor electrode is written as

(15.11)

where e⁻ represents electrons and v represents empty electron states in the conduction band. The concentration of empty states at the interface is high and v can be assumed constant and incorporated into k_a. The forward and reverse reaction rates can be written as follows:

(15.12a)

(15.12b)

The forward (cathodic) reaction depends on the concentration of electrons at the surface, n_s, which depends on the potential. In contrast, with the anodic reaction, electrons are being injected into a nearly empty band. Thus, for n-type semiconductors, the reverse (anodic) reaction depends only on the relative position of the energy levels and, therefore, its rate remains constant with potential when . The potential change is absorbed by the semiconductor as the width of the depletion region changes while the current remains constant. Consequently, the flow of current in the dark for an n-type semiconductor is

(15.13)

where  is the saturation current, and CB refers to the conduction band. A similar expression applies to the valence band for p-type materials:

(15.14)

From Equations 15.13 and 15.14, it is readily apparent that significant current is only able to flow in one direction. This behavior is shown in Figure 15.14 for both p– and n-type electrodes. For the n-type semiconductor, the cathodic current increases exponentially for negative overpotentials, similar to what we have observed for metal–electrolyte systems. In contrast, the current for positive overpotentials is relatively independent of potential and much lower in magnitude for n-type semiconductors. Note that cathodic current is negative, consistent with the definition that we have been using throughout the text.

Similar to the p–n junction of a semiconductor, the junction between an electrolyte and a semiconductor acts like a diode, with rectifying properties. In the terminology of the semiconductor field, when the magnitude of the current is large, the applied potential is called a forward bias. When the magnitude of the current is small, it is described as being under a reverse bias. We can use these terms for the junction between an electrolyte and a semiconductor. However, we must recognize that the behavior of a junction between an electrolyte and a semiconductor changes depending on whether it is n- or p-type. This difference is clearly shown in Figure 15.14. For an n-type semiconductor, large cathodic currents result under a forward bias, which corresponds to a negative overpotential. On the other hand, for a p-type semiconductor, large anodic currents result when a positive overpotential is applied; this also corresponds to a forward bias.

A Tafel plot can be used to determine  or  under forward bias if the equilibrium potential is known. Under the simplifying assumptions considered here, the equilibrium potential is determined by the redox couple in solution and the semiconductor equilibrates by passing charge to or from the solution.

ILLUSTRATION 15.4

An n-type semiconductor electrode operating in the dark has a current density under reverse bias of 6 × 10⁻⁵ A·cm⁻². The equilibrium potential is −1.20 V versus SCE. What are the potential and current density at a forward bias of 0.20 V with respect to the equilibrium potential? Assume 25 °C. Plot the current density versus potential.

SOLUTION:

Forward bias for an n-type semiconductor results from application of a negative potential. Therefore, the voltage is −1.20 V − 0.20 V = −1.40 V, and η = −0.20 V. . Substituting these values into Equation 15.13 yields

where the negative sign indicates that the current is cathodic, consistent with the sign convention that we have been using in this text. The current density versus voltage curve (in the dark) is shown in the figure.

At negative overpotentials (potentials below −1.2 V), the current increases exponentially, similar to what we observed earlier for metal electrodes. In contrast, the current approaches a constant (small) value of  at positive overpotentials for an n-type semiconductor. This curve illustrates the rectifying effect of semiconductors in the absence of illumination.

The language used in semiconductor photoelectrochemistry is a blend of expressions from solid-state physics, the semiconductor industry, and electrochemistry. You should be familiar with these terms.

Forward bias: Similar to p–n junctions of doped semiconductors, the interface between an electrolyte and a semiconductor acts like a diode. The lexicon from electrical engineering is often used to describe the interface between the electrolyte and a semiconductor. Namely, forward bias refers to the condition where the applied potential results in a current of large magnitude. However, bias and overpotential, η, are not the same. Most importantly, the behavior of a semiconductor–electrolyte interface is different depending on the type of doping.

n -type:	η < 0, forward bias, band bending decreases or is reversed, the thickness of the space-charge layer decreases, large cathodic current results
	η > 0, reverse bias, band bending increased, thickness of the space-charge layer increases, very small anodic current
p-type:	η > 0, forward bias, large anodic current
	η < 0, reverse bias, very small cathodic current

Drift current: What an electrochemist would call migration, many electrical engineers refer to as drift. The physics are the same, namely, the movement of charge particles (electrons or holes) in the presence of an electric field.