Now that we have examined the electrochemical behavior of semiconductors in the dark, the next step is to include photoelectrochemical (PEC) effects. Toward that end, this section describes the absorption of light by semiconductors. The principal mechanism for the absorption of photon energy by a semiconductor is the excitation of an electron from the valence to the conduction band. This takes place when the energy of the light exceeds that of the band gap as shown in Figure 15.15. Consequently, there is little absorption of light whose energy is below that of the band gap, and the absorption that does occur is the result of additional energy states that are present owing to chemical impurities or physical defects.

Light is typically characterized by its wavelength (λ) or its frequency (ν). In order to determine if light of a particular wavelength will be absorbed, we need a relationship between the wavelength, frequency, and energy. Since the energy of a photon is hν and ν = c/λ,

(15.15)

The cutoff wavelength, λ_c, for absorption is the wavelength that corresponds to the band gap energy and is found as

(15.16)

Light absorption decreases precipitously for wavelengths greater than the cutoff value defined by Equation 15.16. In advanced photoelectrochemical systems, the photons that are not absorbed in one semiconductor are transmitted to a second semiconducting absorber layer having a lower band gap so that their energy is not wasted.

Light is absorbed gradually as it passes through a semiconductor, as illustrated in Figure 15.16. Therefore, the amount absorbed is a function of the thickness of the semiconductor through which the light is passing. The local rate of absorption is proportional to the flux at that point, described mathematically as

(15.17)

where  is the photon flux per unit area, and the proportionality constant, α, is called the absorption coefficient.

The fraction absorbed as a function of distance from the surface can be obtained from Equation 15.17:

(15.18)

where  is the entering flux (photons s⁻¹ m⁻²) and  is the exiting flux at a distance x from the surface. The entering flux is equal to the incident flux in situations where reflection is not important. Otherwise, it is the difference between the incident flux and the fraction of that flux that is reflected. A larger α corresponds to higher absorption and, therefore, a shorter distance (thinner layer) to absorb a given amount of light. Note that the flux can also be expressed in terms of energy, rather than just photons. In this case, it is typically referred to as irradiance or energy density with units of W·m⁻². As noted above, for a given wavelength, λ, the energy per photon is hc/λ, where h is Planck’s constant, 6.62607 × 10⁻³⁴ J·s. The absorbed flux in units of energy is then

(15.19)

Equation 15.18 applies equally as well for the ratio of energies as it does for that of the photon fluxes.

Absorption coefficients for a few semiconducting materials are shown in Figure 15.17. There are two types of semiconductor band gaps: direct and indirect. With direct band gap materials, the absorption coefficient increases rapidly for photons with an energy greater than E_g, which, for example, is 1.4 eV for GaAs. Light absorption for indirect band gaps requires an additional transition involving the transfer of momentum from the electron to the crystal lattice; that is, in addition to the generation of a hole and electron, a phonon is created. For this reason, the increase in absorption coefficient with photon energy is more gradual for indirect band gap materials such as silicon.

The absorption coefficient for silicon is approximated by the following equation over the range of wavelengths from 400 to 1140 nm:

(15.20)

where λ is in nm and α is in cm⁻¹. Note that errors in the fit are magnified because of the logarithm, and measured data for the conditions of interest are always best.

What happens to photons with energy that are well above the band gap? Electrons are created with energies greater than the conduction band edge. It would be ideal if we could convert this to electrical work; however, most of this excess energy is quickly converted to heat. For some devices, this may not be an issue, but if we are trying to maximize the conversion of solar irradiance to electrical work, this energy is wasted and results in a loss of efficiency. This leads to an important trade-off that affects the overall efficiency of solar devices. Keep in mind that such devices are typically exposed to light with a broad range of wavelengths. A semiconductor with a small band gap can absorb a larger fraction of this light. However, much of the absorbed energy is lost since excess photon energy is dissipated as heat to the semiconductor crystal (see Figure 15.15). As the band gap increases, a smaller fraction of the available photons are absorbed, but the loss of energy as heat is also reduced. The net result is a maximum efficiency for the solar spectrum of about 34% at a band gap of 1.34 as shown in Figure 15.18.

Importantly, silicon has an indirect band gap and light just below the critical wavelength tends to penetrate a significant distance before it is absorbed. Consequently, the thickness of silicon devices (∼100 μm) is significantly greater than that required for direct band gap materials such as CdTe or CIGS (copper indium gallium (di)selenide) (∼1 μm).

ILLUSTRATION 15.5

Light Absorption by Silicon

1. Single-crystal silicon is exposed to monochromatic light at a wavelength of 600 nm. The incident power due to the light is 10 mW. The thickness of the crystal (orthogonal to the incident light) is 0.5 μm. Please determine the total rate of energy absorption by the silicon. You may assume that the 10 mW is the energy that enters the silicon (you do not need to account for reflection at the surface).
2. What is the minimum energy of light that can be absorbed by silicon? To what wavelength does this correspond?
3. If light with energy in excess of the minimum energy is absorbed, what happens to the extra energy?

SOLUTION:

1. We need to determine the fraction of the light that enters the Si crystal that is absorbed in the 0.5 μm thickness of the crystal. The absorption coefficient for crystalline silicon at 600 nm is approximately 3.92 × 10³ cm⁻¹ (see Equation 15.20). From Equation 15.18, the fraction absorbed isTherefore, the rate of energy absorption is 1.78 mW.
2. The minimum energy that can be absorbed is equal to the band gap energy. From Figure 15.17 for crystalline Si, E_g ≈ 1.1 eV, which is seen by the sharp increase in absorption for energies higher than this value. The corresponding wavelength can be determined from Equation 15.15:
3. Energy in excess of the band gap goes to heat, which is dissipated into the crystal.