Energy Scales

Before going any further, we need to relate the energy scale commonly used with semiconductors to the standard hydrogen electrode (SHE) scale familiar to electrochemists and electrochemical engineers. As you know, the SHE scale uses the hydrogen electrode under standard conditions as the reference potential for all electrochemical reactions. In contrast, the most common reference state used for semiconductors is an electron energy equal to zero in a complete vacuum (Evacuum = 0). On this vacuum scale, the energy of an electron bound in a solid (expressed in electron volt [eV]) is typically negative since its energy is lower than that of a free electron in a vacuum. The SHE potential scale can be converted to an energy scale (eV) by multiplying by the charge on an electron. In doing so, one finds that the scale increment is the same for both scales (see Illustration 15.2) and that the hydrogen scale is simply shifted relative to the vacuum scale. Specifically,

(15.2)
As we discussed earlier in Chapter 3, redox couples with positive potentials relative to hydrogen have lower electron energies. Therefore, positive values of yield energies more negative than −4.44 eV on the vacuum scale. Figure 15.9 shows the band gap energy and positions of the band edges (EC and EV) for several different semiconductors on (i) the vacuum scale and (ii) the SHE scale.

Figure 15.9 Band gap energies at 300 K as well as location of the band edges for various semiconductors. Both vacuum and hydrogen scales are shown. Solids with band gaps larger than about 3 eV are effectively insulators.

Energy levels are also important. The Fermi level, EF, in a semiconductor is defined as the hypothetical energy level at which it is equally probable that the level is either occupied by an electron or vacant. In addition to this probabilistic definition, thermodynamically we can think of the Fermi level as the electrochemical potential of an electron in the solid. For an intrinsic (undoped) semiconductor, EF lies in the middle of the band gap. In contrast, the presence of donor states in an n-doped semiconductor increases the probability that electrons exist at a higher energy state, resulting in an increase in EF such that it lies slightly below Ec (about 0.1 eV for typical doping levels). Similarly, the presence of low-energy acceptor states in p-type semiconductors means that EF is located just above EV. The relative energies of a redox couple in solution and the Fermi level of the semiconductor determine what will happen when the semiconductor is placed in solution, as discussed in the next section.

ILLUSTRATION 15.2
Energy Units and Voltage

Although it may surprise you, an electron volt is actually an energy unit rather than a voltage (V or J·C−1). Specifically, 1 eV is equal to 1.602 × 10−19 J and is, by definition, the amount of energy gained (or lost) as the charge of a single electron is moved through an electric potential difference of 1 V.

You may be wondering how we can put energy in eV on the same scale as potential (V versus SHE). To get the difference in energy [eV] that is comparable to a specified change in potential [V], we start with a voltage difference and then multiply by the charge on an electron. For a voltage difference of 1 V, it follows:

Therefore, a change in voltage of 1 V on the hydrogen scale is numerically equivalent to an energy change of 1 eV. In other words, the increment for both the vacuum scale [eV] and the potential scale [V] is the same. The difference between the scales is that the potential scale is shifted so that it has a zero value that corresponds to hydrogen, whereas the zero value on the vacuum scale is the energy of an electron in a vacuum.