1 Introduction

In this chapter we deal with the simplest ideas that have been used in the past to attain an understanding of solar cell efficiencies from a theoretical point of view. The first and most obvious attack on this problem is to use thermodynamics, and we offer four such estimates in Section 2. Only the first of these is the famous Carnot efficiency. The other three demonstrate that one has more possibilities even within the framework of thermodynamics. To make progress, however, one has to introduce at least one solid-state characteristic, and the obvious one is the energy gap, E_g. That this represents an advance in the direction of a more realistic model is obvious, but it is also indicated by the fact that the efficiency now calculated is lower than the (unrealistically high) thermodynamic efficiencies (Section 3). In order to get closer to reality, we introduce in Section 4 the fact that the radiation is effectively reduced from the normal blackbody value (Equation (6)) owing to the finite size of the solar disc. This still leaves important special design features such as the number of series-connected tandem cells and higher-order impact ionisation, and these are noted in Section 5.

2 Thermodynamic Efficiencies

The formulae for ideal efficiencies of solar cells are simplest when based on purely thermodynamic arguments. We here offer four of these: they involve only (absolute) temperatures:

From these temperatures, we form the following efficiencies [1]:

(1)

(2)

(3)

(4)

In the latter efficiency, the cell temperature is determined by the quintic equation

(5)

The names associated with these efficiencies are not historically strictly correct: for example, in Equations (2) and (3) other authors have played a significant part.

Figure 1 [1] shows curves of the four efficiencies, which all start at unity when T_a/T_s≡0, and they all end at zero when T_a = T_s. No efficiency ever beats the Carnot efficiency, of course, in accordance with the rules of thermodynamics. Values near T_s = 5760–5770 K seem to give the best agreement with the observed solar spectrum and the total energy received on Earth, but a less accurate but more convenient value of T_s = 6000 K is also commonly used. Using the latter value of T_s and T_a = 300 K as the temperature for Earth, one finds

If T_s = T_a = T_c one has in effect an equilibrium situation, so that the theoretical efficiencies are expected to vanish.

The above thermodynamic efficiencies utilise merely temperatures, and they lie well above experimental results. One needs an energy gap (E_g) as well to take us from pure thermodynamics to solid-state physics. Incident photons can excite electrons across this gap, thus enabling the solar cell to produce an electric current as the electrons drop back again. The thermodynamic results presented earlier, on the other hand, are obtained simply by considering energy and entropy fluxes.

3 Efficiencies in Terms of Energies

In order to proceed, we need next an expression for the number of photons in blackbody radiation with an energy in excess of the energy gap, E_g say, so that they can excite electrons across the gap. At blackbody temperature T_s the number of photons incident on unit area in unit time is given by standard theory as an integral over the photon energy [2]:

(6)

Now suppose that each of these photons contributes only an energy equal to the energy gap to the output of the device, i.e., a quantity proportional to

(7)

To obtain the efficiency η of energy conversion, we must divide this quantity by the whole energy that is, in principle, available from the radiation:

(8)

Equation (8) gives the first of the Shockley–Queisser estimates for the limiting efficiency of a solar cell, the ultimate efficiency (see Figure 5). The argument neglects recombination in the semiconductor device, even radiative recombination, which is always present (a substance that absorbs radiation can always emit it!). It is also based on the blackbody photon flux (Equation (6)) rather than on a more realistic spectrum incident on Earth.

We shall return to these points in Section 4, but first a brief discussion of Equation (8) is in order. There is a maximum value of η for some energy gap that may be seen by noting that η = 0 for both x_g = 0 and for x_g very large. So there is a maximum efficiency between these values. Differentiating η with respect to x_g and equating to zero, the condition for a maximum is

corresponding to η = 44%.

This is still higher than most experimental efficiencies, but the beauty of it is that it is a rather general result that assumes merely properties of blackbody radiation.

Let f(x) be a generalised photon distribution function; then a generalised efficiency can be defined by

(9)

The maximum efficiency with respect to x_g is then given by

(10)

This is rather general and will serve also when the photon distribution departs from the blackbody forms and even for radiation in different numbers of dimensions.

4 Efficiencies Using the Shockley Solar Cell Equation

A further step in finding the appropriate efficiency limits for single-junction solar cells can be made by estimating the relevant terms in the Shockley ideal solar cell equation (Equation (1) in Chapter Ia-1). To this end, further remarks must be made about the solar spectrum and solar energy incident on Earth’s surface. The ultimate efficiency, discussed in Section 3, was based on the blackbody photon flux (Equation (6)), a rigorous thermodynamic quantity but not a very good estimate of the solar spectrum as seen on Earth. By virtue of the large distance between the Sun and Earth, the radiative energy incident on Earth’s surface is less than that of Equation (6) by a factor fω, which describes the size of the solar disk (of solid angle ω_s) as perceived from Earth:

(11)

where R_s is the radius of the Sun (696×10³ km), and R_SE is the mean distance between the Sun and Earth (149.6×10⁶ km), giving ω_s = 6.85×10⁻⁵ sterad and fω = 2.18×10⁻⁵. The resulting spectrum is shown in Figure 2 alongside the standard terrestrial AM1.5 spectrum (a further discussion of the spectra that are used for solar cell measurements in practice can be found in Chapter III-2, which also shows the extraterrestrial spectrum AMO).

The maximum value of the photogenerated current I_ph now follows if we assume that one absorbed photon contributes exactly one electron to the current in the external circuit:

(12)

where A is the illuminated area of the solar cell and q is the electron charge. The maximum photogenerated current density J_ph = I_Ph/A that can be produced by a solar cell with band gap E_g is shown in Figure 3. To allow a comparison with photocurrents measured in actual devices, Figure 3 is plotted for the AM1.5 solar spectrum, which is used for calibration of terrestrial solar cells, rather than for the blackbody spectrum used in Section 3.

The open-circuit voltage V_oc can now be obtained using the photogenerated current I_ph (Equation (12)) and the (dark) saturation current I₀ that appears in the ideal solar cell equation:

(13)

The current I₀ can be obtained by a similar argument as the photogenerated current I_ph, since, as argued by Shockley and Queisser, it can be equated to the blackbody photon flux at the cell temperature T_c (in what follows, the cell temperature T_c will be assumed to be equal to the ambient temperature T_a):

(14)

where the coefficient f₀ has been inserted to describe correctly the total area f₀A exposed to the ambient photon flux. Various values of f₀ (some dependent on the refractive index n of the cell material) can be found, appropriate for different device structures and geometries. The usual value is f₀ = 2, as suggested by Shockley and Queisser [2], since this radiation is incident through the two (front and rear) surfaces of the cell. A similar argument for a spherical solar cell yields an effective value f₀ = 4 [3]. Henry [4] gives f₀ = 1+n² for a cell grown on a semiconductor substrate, but the value f₀ = 1 is also sometimes used (see, for example, [5]). Green [6] gives a semi-empirical expression for the dark saturation current density J₀ = I_o/A:

(15)

An approximate analytical method for estimating V_oc can also be useful, particularly as it stresses the thermodynamic origin of V_oc. Indeed, it can be shown [7] that, near the open circuit, the solar cell behaves as an ideal thermodynamic engine with Carnot efficiency (1−T_c/T_s). Ruppel and Würfel [3] and Araùjo [8] show that V_oc can be approximated to a reasonable accuracy by the expression

(16)

which depicts the dependence of V_oc on the band gap E_g and on the cell temperature T_c. Figure 4 compares this theoretical values for the open-circuit voltage with data for the best solar cells to date from different materials.

Using now an expression for the fill factor (defined by Equation (3) in Chapter Ia-1), one readily obtains a theoretical estimate for the efficiency. Slightly different results may be encountered, principally by virtue of the different ways one can estimate the current and the voltage. Figure 5 shows the best-known result, the celebrated Shockley–Queisser ideal efficiency limit [2]. Shockley and Queisser call this limit the nominalefficiency, to be compared with the ultimate efficiency, which is discussed in Section 3. Figure 5 shows two such curves: one labelled ‘one-sun’ corresponds to the AMO solar intensity, as observed outside Earth’s atmosphere. A second curve, labelled ‘maximum concentration’ corresponds to light focused on the cell, by a mirror or a lens, at the maximum concentration ratio of l/fω = 45,872 [9].

The various unavoidable losses in photovoltaic energy conversion by single-junction solar cells can be depicted in a graph constructed by Henry [4] and analogous to Figure 6. There are two curves in this graph:

The three shaded areas then depict the three fundamental losses in a single junction solar cell (shown here for silicon with band gap E_g equal to 1.12 eV):

The area of the blank rectangle then represents the maximum efficiency that can be obtained for a single junction cell made from a semiconductor with band gap E_g. The graph is drawn here for light with maximum possible concentration. A different ‘voltage curve’ would result if light with one-sun intensity were used.

5 General Comments on Efficiencies

The ideal solar cell efficiencies discussed above refer to single-junction semiconductor devices. The limitations considered in the ultimate efficiency of Section 3 are due to the fact that the simplest semiconductor (i.e., one whose defects and impurities can be ignored) cannot absorb below-band-gap photons. Furthermore, it is also due to the fact that the part of the energy of the absorbed photons in excess of the band gap is lost as heat. Radiative recombination at the necessary fundamental level was taken into account in the treatment of Section 4. It is sometimes argued that there are other ‘unavoidable’ losses due to electronic energy transfer to other electrons by the Auger effect (electron–electron collisions) [10–12]. There is also the effect of band-gap shrinkage, discussed in Chapter Ia-2, and light trapping may also play a part [11]. None of these effects are discussed here, and the reader is referred to the relevant literature.

It is clear that it is most beneficial if one can improve the effect of a typical photon on the electron and hole density. This can be achieved, for example, if the photon is energetic enough to produce two or more electron–hole pairs. This is called impact ionisation and has been studied quite extensively. A very energetic photon can also project an electron high enough in to the conduction band so that it can, by collision, excite a second electron from the valence band. This also improves the performance of the cell. On the other hand, an electron can combine with a hole, and the energy release can be used to excite a conduction-band electron higher into the band. In this case, energy is uselessly dissipated with a loss of useful carriers and hence of conversion efficiency. This is one type of Auger effect. For a survey of these and related effects, see [12]. These phenomena suggest a number of interesting design problems. For example, is there a way of limiting the deleterious results of Auger recombination [13]? One way is to try to ‘tune’ the split-off and the fundamental band gaps appropriately. If one is dealing with parabolic bands, then the obvious way is to examine the threshold energies that an electron has to have in order to jump across the gap and make these large so as to make this jump difficult.

Then there is the possibility of placing impurities on the energy band scale in such a way as to help better use to be made of low-energy photons, so that they can now increase the density of electrons in the system. This is sometimes referred to as the impurity photovoltaic effect. So one can make use of it [14].

One can also utilise excitons to improve the efficiencies of solar cells. There may be as many as 10¹⁷ cm⁻³ excitons in silicon at room temperature. If they are split up in the field of a p–n junction, this will increase the concentration of current carriers and so increase the light generated current, which is of course beneficial.

We have here indicated some useful ideas for improving solar cells, There are of course many others, some of which are discussed in Chapters Ib-2 and Id-2 as well as elsewhere [15]. Note, in particular, the idea of developing tandem cells in which photons hit a large band-gap material first and then proceed gradually to smaller band-gap materials. Tandem cells are now available with three of more stages. Solar cells with efficiency of order 20% arc predicted to be produced on a large scale in the near future [16]. Table 2 shows the best laboratory efficiencies at the present time for different materials.

TABLE 1 The Maximum Efficiencies of Tandem Cells as a Function of the Number of Cells in the Stack for Different Concentration Ratios [17]. Note that de Vos [17] Uses a Slightly Smaller Value of fω than Shockley and Queisser, Resulting in a Marginally Different Maximum Concentration Ratio than used in Figure 5

TABLE 2 The Best Reported Efficiencies, at Time of Writing, for Different Types of Solar Cells [18]

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