The suppositions laid out in Section 3.4.1 are extremely rough and are based solely on the principle of photon absorption in semiconductors, but not on the other relevant aspects of semiconductor physics that come into play here. Far more realistic results can be obtained if it is also assumed that the electric field in the barrier layer of a semiconductor diode is used to separate the photon-generated electron-hole pair. The simplified equivalent circuit of Figure 3.12 is used for this purpose, but without RS and RP. This method allows for the calculation of theoretical efficiency цт in an idealized solar cell as a function of band gap energy Eg, based on spectral efficiency zs and using Equations 3.6, 3.11 and 3.12. This method duly takes account of the key laws of semiconductor physics, while still leaving intact the following idealized suppositions: 
This method takes account of two key factors: (a) the solar cell current VMpp in terms of its MPP will be far lower than the theoretical photovoltage Vph — EG/e; and (b) the current lMpp at the MPP will be lower than lSC.
Equation 3.11 realized for Pmax yields the following in this case:
where FFj – is the idealized fill factor in accordance with Equation 3.12 and SF — VOC/Vph is the voltage factor.
Hence the following applies to theoretical efficiency zt:
It therefore follows that lSC/AZ equates to the maximum current density JSCmax — Jmax in Equation 3.18.
Hence, to achieve maximum efficiency zt, open-circuit voltage Voc and thus voltage factor SF need to be as high as possible. According to Equation 3.6, saturation current lS needs to be as low as possible for a high open-circuit voltage Voc. The minimum possible saturation current density, which is JS — lS/AZ, can be estimated using the following equation [Gre86]:
Vph — EG/e, i. e. theoretical photovoltage according to Equation 3.16 VT — nkT/e, i. e. thermal diode voltage according to Equation 3.4
According to [Gre86], in order to obtain a reasonably accurate estimate of the minimum possible JS, it is necessary to apply something along the lines of Ks — 150000 A/cm2. For a silicon solar cell where Eg — 1.12eV, a value of around 18fA/cm2 for JS is obtained with this method at 25°C. More recent findings show that a minimum of JS ~ 5 fA/cm2 may even be attainable [3.1]. This results in a Ks value of around 40 000 A/cm2.
Inasmuch as lph/lS ~ lSC/lS — JSC/JS, the JS used for Equation 3.21 can be applied to Equation 3.6, thus resulting in a very simple equation for maximum possible open-circuit voltage Voc:
Based on the values used for Equation 3.15 and those indicated for JSCmax — Jmax in Figure 3.20, using Equation 3.6 or 3.22, the maximum possible open-circuit voltage Voc and the voltage factor SF in a solar cell can now be determined as a function of EG, for diode quality factor n — 1:
Figure 3.23 shows the maximum values determined for open-circuit voltage VOC and the voltage factor SF in a solar cell as a function of EG (for diode quality factor n — 1) at STC (1 kW/m2, AM1.5, 25 °C), based on the above suppositions.
Maximum Open Circuit Voltage and Voltage Factor SF
Figure 3.23 Maximum open-circuit voltage VOC and voltage factor SF for an idealized solar cell where n = 1, as a function of semiconductor material band gap energy EG
By the same token, using the maximum Voc F; can be
calculated. Figure 3.24 shows FF; as a function of EG for n = 1 at STC (1 kW/m2, AM1.5, 25 °C).
On account of the logarithmic dependency that comes into play here, with 1 kW/m2 irradiance the voltage factor SF and the idealized fill factor FF; are determined to only a minor extent by spectrum type (AM0 or AM1.5). As can be readily seen in Equations 3.22 and 3.12, both Voc and FF; decrease as the diode quality factor n increases, which means that optimally low n values close to 1 yield the best results.
Idealized Fill Factor FF; of a Solar Cell with n = 1
Figure 3.24 Idealized fill factor FF; as for Figure 3.23, determined using Equation 3.12 for a solar cell where n = 1, as a function of semiconductor material band gap energy EG
Figure 3.25 Theoretical efficiency zt of an ideal solar cell as a function of band gap energy EG for semiconductor material for n = 1 in the AM1.5 spectrum with 1 kW/m2 and in the AM0 spectrum
Using Equation 3.20 and multiplying the curves in Figures 3.21, 3.23 and 3.24, the theoretical efficiency zt of a solar cell for the AM0 and AM1.5 spectra with 1kW/m2 can now be obtained. Figure 3.25 shows zt as a function of band gap energy EG of solar cell material with 1 kW/m2 and a cell temperature of 25 °C. As with spectral efficiency zS, theoretical efficiency zt for the AM1.5 spectrum is several percentage points higher than for the AM0 spectrum, owing to somewhat lower infrared content.
Hence semiconductor materials whose EG ranges from around 0.8 to 2.1 eVare particularly well suited for use in solar cells. The roughly 1.1 to 1.6 eV range is particularly suitable by virtue of the fact that theoretical efficiency zt in the AM1.5 spectrum is likeley to be upwards of 28%. Indium phosphide (InP) and gallium arsenide (GaAs) have almost exactly the right EG values for such applications, but crystalline silicon also works very well by virtue of its 1.12 eV and zt of 28.5%.
A comparison of the theoretical efficiency zt curve in the AM1.5 spectrum and spectral efficiency zs as in Figure 3.21 reveals a substantial difference between the two curves, particularly for low EG:
For c-Si: Eg « 1.12 V, zS ~ 48%, zt ~ 28.5%.
For Ge: Eg « 0.66 V, zs ~ 40%, zt ~ 13.5%.
In view of the fact that (as noted in Section 3.2.2) (a) diffusion voltage VD at ambient temperature is roughly 0.35 to 0.5 V lower than the theoretical photovoltage of 1.12 Vand 0.66 V, (b) open-circuit voltage VOC is somewhat lower than VD, and (c) solar cell voltage VMPP at the MPP is somewhat lower than VOC, the values indicated are extremely illuminating.