*3.4.3. The efficiency calculation*

Energy conversion efficiency *η* is usually known as the most relevant figure for solar cell performance. Solar cell efficiency is calculated by dividing a cell's electrical power output at its maximum power point by the input solar radiation and the surface area of the solar cell.

**Figure 12.** The *Voc/Eg* against energy band-gap of the solar cell, using a blackbody spectrum at *Ts*=6000°*K*; a)-for differ‐ ent radiative recombination rates, b)-for different solar concentrations and a plot of (56) for *C=Cmax*.

The maximum power output from the solar cell is obtained by choosing the voltage *V* so that the product current-voltage (*IV*) is a maximum. This point corresponds to the situation where a maximum power is extracted from the cell. Using equation 45 we can define the power delivered by a cell as:

$$P(V) = (q \gets f \phi\_s + q(1 - \mathbb{C} \, f)\phi\_a - \frac{q}{f\_{RR}}\phi\_c(V)) \times V \tag{58}$$

The maximum power *PM* = *JM* ×*VM* is obtained numerically from (58) and the efficiency *η* (if the sun is assumed a blackbody at *Ts*) is then calculated as:

$$\eta(\%) = \frac{P\_M}{P\_{\rm in}} = \frac{P\_M}{2\pi^5 \left(kT\_s\right)^4 / 15h^3c^2} \times 100\tag{59}$$

For AM1.5G solar spectrum *Pin* is replaced by 1000.

It is worth mentioning that (56) is not an exact evaluation of *Voc*. As shown in figure 12.b equation (56) for narrow band-gap semiconductors yields wrong values of *Voc,max* (above *Eg/q* line), acceptable values are obtained only for *Eg* greater than 2 eV, where it coincides with the

From this figure one can say that taking *Voc,max*=*Eg/q* is a much better approximation; thus,

max <sup>1</sup> ln ln *<sup>g</sup> c c*

æ ö æ ö =- - ç ÷ ç ÷ ç ÷ è ø è ø

A more accurate value of *Voc* is obtained after numerical resolution of equation (45) for *J*(*V*)=0,

The other type of entropy loss degrading the open-circuit voltage is the photon entropy increase due to isotropic emission under direct sunlight. This entropy increase occurs because solar cells generally emit into 2π steradian, while the solid angle subtended by the sun is only

The most common approach to addressing photon entropy is a concentrator system. If the concentration factor *C* of sun radiation is increased, this is generally achieved by optical means, a significant increase of *Voc* is obtained (as shown in figure 12.b). The calculation is carried out assuming a dominant radiative recombination and with maximum external fluorescence efficiency (*fRRηfex*=1). In this case we can notice that as *C* is increased *Voc* approaches its ultimate value *Eg*/*q*, for example GaAs (*Eg*=1.43 eV) for *C=CMax* we get *Voc*=0.99×*Eg*/*q*=1.41 V, which corresponds to an efficiency limit of approximately 38.5% (blackbody spectrum at 6000*K*). This theoretical limit shows the importance of dealing with entropy losses associated with angle of acceptance of photons from the sun and emission of photons from the cell efficiently. This value is well above the predicted SQ limit, where the concentration factor was considered. With reference to table 1 we can clearly see that the record open circuit voltage under one-sun condition (*C*=1) of gallium arsenide solar cell (1.12 *V*) is already close to the SQ limit (1.17 *V*) while silicon solar cell is still behind with a record *Voc* of 0.706 *V* compared to a limit of 0.893 *V*, this difference is due to the fact that GaAs has a direct band gap, which means that it can be used to absorb and emit light efficiently, whereas Silicon has an indirect band gap and so is relatively poor at emitting light. Although Silicon makes an excellent solar cell, its internal fluorescence yield is less than 20%, which prevents Silicon from approaching the SQ limit [20]. On the other hand It has been demonstrated that efficiency in Si solar cells is limited by Auger

Energy conversion efficiency *η* is usually known as the most relevant figure for solar cell performance. Solar cell efficiency is calculated by dividing a cell's electrical power output at its maximum power point by the input solar radiation and the surface area of the solar cell.

*qq C q f*

*<sup>E</sup> kT C kT <sup>V</sup>*

*RR fex*

h

(57)

result obtained when solving numerically (45) for *J*(*V*)=0.

instead of (56) we can use the following approximation:

recombination, rather than by radiative recombination [20-22].

*3.4.3. The efficiency calculation*

*oc*

70 Solar Cells - New Approaches and Reviews

the results are plotted in figure12a and 12.b.

6.85×10−5 steradian.

Figure 13 illustrates efficiency against energy band-gap of a solar cell, using the AM1.5G spectrum and the blackbody spectrum at *Ts*=6000°*K* for one sun and full concentration (*C=CMax*), the only recombination mechanism is radiative and 100% external fluorescence efficiency, which means that all emitted photons from the cell (issue from radiative recombi‐ nation) are allowed to escape. The maximum efficiency is 34.42% for AM1.5G corresponding to a gap of 1.34 eV, for a blackbody spectrum normalized to 1000 W/m2 the maximum efficiency is 31.22% at 1.29 eV, while for a full solar concentration the maximum is 40.60% at 1.11 eV, this confirms the fact that the optimal band gaps decrease as the solar concentration increases.

In figure 14 the product of radiative recombination rate and the external fluorescence efficiency ( *f RR* ×*η fex*) is decreased from 1 to 10-3 to illustrate the effect radiative recombination and the external fluorescence on the cell efficiency. The maximum efficiency limit dropped from 34.42% to 28.58%.

**Figure 13.** The maximum efficiency against the energy band-gap of the solar cell, using the AM1.5G spectrum with the blackbody spectrum at *Ts*=6000°*K* for one sun and full concentration (*C=CMax*).

Since the power output of the cell is determined by the product of the current and voltage, it is therefore imperative to understand what material properties (and solar cell geometries) boost these two parameters. Certainly, the short-circuit current in the solar cell is determined entirely by both the material absorption property and the effectiveness of photo-generated carriers collection at contacts. As previously mentioned (section 2.4.1), the manufactured solar cells with present technologies and materials have already achieved short-circuit currents close to predicted limits. Therefore the shortfall in efficiency could be attributed to the voltage. We show here that the key to reaching the highest possible voltages is first to have a recombination predominantly radiative with a maximal external emission of photons from the surface of the solar cell. Secondly we need a maximum solar concentration. The second condition could be achieved either by using sun concentrators, there are concentrators with concentration factor from ×2 to over ×1000 [23] or by non-concentrating techniques with emission and acceptance angle limited to a narrow range around the sun [24-26].

At this level we can conclude that the efficiency limit of a single energy gap solar cell is bound by two intrinsic limitations; the first is the spectral mismatch with the solar spectrum which retains at least 50% of the available solar energy. The best known example of how

**Figure 14.** The maximum efficiency against the energy band-gap of the solar cell, using the AM1.5G spectrum with the blackbody spectrum at *Ts*=6000°*K* for one sun and full concentration (*C=CMax*)

to surmount such efficiency restraint is the use of tandem or stacked cells. This alterna‐ tive will become increasingly feasible with the likely evolution of materials technology over the decades to 2020 [27].

The second intrinsic loss is due to the entropy associated with spontaneous emission. To overcame this limitation three conditions should be satisfied, that is: a) – prevailing radiative recombination (to eliminate the non-collected electro-hole pairs), b)-efficient external fluores‐ cence (to maximise the external emission of photons from the solar cell) and c)-using concen‐ trated sun light or restricting the emission and acceptance angle of the luminescent photons to a narrow range around the sun.
