Some crude estimates may be useful. For example, the open-circuit voltage has an upper limit Eg/e, where Eg is the semiconductor bandgap. The analysis  assumes a single junction and a single uniform bandgap energy. It is assumed that photons whose energy is less than the bandgap energy Eg do not contribute to the photocurrent. It is also assumed that the excess energy hc/l — Eg is lost to heat in the semiconductor. In practical terms, the junction structure should be thick enough that all of the light of energy hc/l — Eg > 0 is absorbed.
A disadvantage of the single-junction cell at bandgap Eg is that all photons of energy less than Eg are lost, are not absorbed but pass through the cell. The energy output of the cell is the same for N photons of 2 Eg or 3 Eg as for N photons of energy Eg. For these reasons, the efficiency of the single-junction cell is inherently limited and depends on the spectrum of photon energies that are incident. Roughly, the bandgap for best efficiency, within the single-junction assumption, should be close to the peak of the incident spectrum.
Recalling Eq. 5.1 and Figure 5.2 the light spectrum from the sun is approximately that of a black body of temperature 5973 K, which peaks at 1m = (2.9 x 106/5973) nm = 486 nm, which corresponds to an energy 1240/
486 = 2.55 eV. (The spectrum at ground level (“Air Mass 1”) is substantially altered, by strong absorptions from several minor constituents of the atmosphere, including ozone, water vapor, and compounds ofnitrogen and carbon. It is also weakened and redshifted by scattering in the atmosphere.) Adopting the hypothetical black body spectrum, for simplicity, if the bandgap of silicon is 1.12 eV, then the many photons in the range below 1.12 eV are completely lost, and an increasing fraction of the energy of the more energetic photons is also lost. The most probable photon at energy 2.55 eV will contribute not more than will a 1.12 eV photon, the difference energy 1.43 eV contributing only to heat.
The specialists in photovoltaic conversion have adopted an effective spectrum “Air Mass 1.5 G,” correcting for an average daytime light path length 1.5 larger than when
provided by adding a highly doped N+-GaAs layer. Light absorption in the rear N-layer is enhanced by the BSF, which inhibits recombination of minority holes with majority electrons at the rear surface. These diagrams makeclearthatlightabsorptionprimarilyoccurs in the field-free regions ofdimensions Lp and Ln beyond the junction-forming depletion layer (, Figure 2.4, p. 25).
the sun is directly overhead. This spectrum is reduced from the top-of-the-atmo – sphere spectrum, Air Mass 0 (AM 0), byabout 28%, ofwhich 18% is from absorption and 10% from scattering. Scattering is well known to follow a Я-4 law, which removes blue light from the direct path but adds in some blue light scattered from the blue sky. The resulting AM 1.5G spectrum (which also includes a diffuse light contribution) is used to calculate the ideal conversion efficiency of a single-junction solar cell operating at 300 K, as a function of the semiconductor bandgap energy. The results for the AM 1.5 G spectrum (labeled, in Figure 6.8) peaks near 29% efficiency at a
means sun spectrum in space, while “AM 1.5” is agreed test spectrum representing noon on a sunny day at a medium latitude, so the sun is not directly overhead at noon and the air traversal path length is about 1.5 the vertical height ofthe air mass. Dashed line represents best efficiency that could be obtained with fictitious black body sun spectrum 6000 K, while AM 0 is a measured spectrum above earth’s atmosphere .
bandgap in the vicinity of 1.4 eV. This is close to the bandgap energy for GaAs, as is shown in Figure 6.8.
The efficiency of the single-junction cell, around 30% for one sun intensity, was addressed by Shockley and Quiesser .
The starting point is the diagram ofthe bands in the p-n junction, shown in Figure 3.16c. If we imagine the sketched open-circuit PN junction cell in thermal equilibrium at a temperature T we realize that dynamical processes ofthermal generation of electron-hole pairs occur throughout the structure to maintain the equilibrium distributions of electrons and holes in the various regions of the junction structure. The reverse current density Jrev comes from diffusion of minority carriers located within the diffusion lengths L of the junction, which carriers fall down the potential gradient. Again, electrons in a p-type semiconductor are of limited lifetime, since they can fall into an ionized acceptor site, give off light, and disappear. The statistics require a corresponding generation process to maintain the equilibrium concentration. The distance the minority electron in the p-region can diffuse before recombination is the minority carrier diffusion length L. Only carriers within such a distance of the actual junction can be usefully driven around the external circuit. At open circuit, a small forward bias appears, such that the net current across the junction is zero. The generation processes in equilibrium are balanced by recombination of electrons and holes to create photons. The recombination events can create photons of energy equal to or larger than Eg = hc/l, where l is the wavelength of the photon.
The insight of Shockley and Quiesser was to recognize that these internal processes must support emission of a black body spectrum, at least for wavelengths shorter than hc/EG from this open-circuited thermal equilibrium structure. The amount of light emitted, fixed by Planck’s radiation law, is closely related to the reverse current density, Jrev, and the spectrum of the light, required by Planck’s radiation law, will be related to the energy distributions of excited electrons and holes in the thermal equilibrium structure. If we recall Figure 5.1, and imagine the small black area A at the bottom of the figure to be the open-circuit solar cell at temperature T in the dark. It will act as a black body and radiate power outward, and the two cases of interest are with and without the concentrating optical system. In the absence of the optical system, a tiny fraction of the black body radiation will fall into an angular range comparable to that of the sun seen from earth, f = 2.16 x 10—5. With the optical system in place, all of the black body radiation will be focused into a solid angle comparable to that of the sun as seen from earth. According to Planck’s law, the total power emitted by the junction of area A into the full hemispherical range of angles, solid angle = 2p, in the photon energy range E1-E2, is, writing power P = dE/dt = E’, based on Eq. 1.1 multiplied by c/4,
where E1 = Eg. The upper limit on the energy may be approximated as large, E2 = 1, or may be estimated from the band structure ofthe semiconductor. The point is this completely determined number of watts W comes from band-to-band recombination in the solar cell, and gives a number that can be closely related  to Jrev (3.66)
AJrev = І0 = eA(2pkT/h3c2)[EG + 2kT Eg + 2(kT)2]exp(—Ec/kT). (6.8)
It is also true that if the junction at temperature T acquires a voltage V between its terminals, then Equation 6.8 is simply multiplied by exp(eV/kT). We now write the current I in terms of the rate of change
of photons exchanged between the area A junction at temperature Tc and the sun at temperature Ts, as in Figure 5.2.
I = Aefs N'(Eg, E2 = i, Ts) — Aefc N'(Eg, E2 = 1, Tc) [exp(eV/kT)-1] Amperes.
Set fs = 2.16 x 10—5 to describe the absence of concentrating optics, and f =1 to describe the area A radiating into all hemispherical directions 2p. The first term is the black body spectrum of the sun at Ts in the energy range above EG. (See Equations 1.1 and 5.1 and related discussion.) The second term is the returning black body radiation of the cell at T = Tc at its chosen open-circuit voltage V. We are describing an open-circuit cell, so I = 0. Taking the sun at 6000 K and the cell at 300 K, Shockley and Quiesser  thus found the best efficiency is 31% for EG = 1.3 eV. The
uppermost dashed curve in Figure 6.8 describes this result, using the vacuum given by Planck’s law of solar spectrum.
Changing to the concentrating situation, where both f factors are 1.0 (input and output radiation both use a full hemispherical range of angles), the new equation is
I — Ae N'(Eg, E2 — i, Ts) — AeN'(EG, E2 — i, Tc) [exp(eV/kT)—1] Amperes.
It is now seen that a larger open-circuit voltage V is needed to balance the larger solar input from the concentrating optics. This means a larger efficiency, since the open- circuit voltage times the current is the power. In this case, Shockley and Quiesser thus find efficiency 40.8% at 1.1 eV for a fully concentrated single-junction solar cell; see Figure 5.1 for an example of such optics, where the imagined cell of area A is at the bottom of the field.
This analysis is based on the direct sunlight and does not include correction for the diffuse scattered light. This analysis is invalid for a cloudy sky.
In this way, the complex situation was analyzed to provide the maximum efficiency of the single-junction device as a function of the single bandgap energy, under assumed illumination conditions. The solid line plots of Figure 6.8 are numerically obtained from the Shockley-Quiesser analysis using the below-atmosphere spectra, as illustrated in Figure 5.2.
As we will see later, the most straightforward means of improving the solar cell efficiency is to put two or more single-gap cells in tandem (series connection), so that the highest energy photons are processed with the largest bandgap junction, and later cells process those photons whose energy was insufficient to generate electron-hole pairs in the prior junctions. A cascade of tandem cells can approach the efficiency of the Carnot machine, in principle. In practice, tandem cells of at least 40% efficiency under concentration have been demonstrated. The materials science and engineering of these tandem structures make them more expensive, but this can be counterbalanced by using concentrating light systems. The cells are more efficient at higher light intensity, because the open-circuit voltage increases with light intensity.