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Theoretical Limiting Efficiency

We can estimate the limiting efficiency of ideal infinite multi-junction solar cells using the graphical quantum-efficiency (QE) analysis invented by C. H. Henry.1 To fully take advantage of Henry’s method, the unit of the AM1.5 spectral irradiance should be converted to that of photon flux (i.e., number of photons/m2/s). To do that, it is necessary to carry out an intermediate unit conversion from the power of electromagnetic radiation incident per unit area per photon energy to the photon flux per photon energy (i.e., from [W/m2/eV] to [number of photons/m 2 /s/eV]). For this intermediate unit conversion, the following points have to be considered: A photon has a distinct energy which is defined as follows.

(1): Eph = h∙f = h∙(c/λ)

where Eph is photon energy, h is Plank’s constant (h = 6.626*10-34 [J∙s]), c is speed of light (c = 2.998*108 [m/s]), f is frequency [1/s], and λ is wavelength [nm].

Then the photon flux per photon energy, dnph/dhν, with respect to certain irradiance E [W/m2/eV] can be calculated as follows.

(2): $$\frac{dn_{ph}}{dhv} \, = \frac{E}{E_{ph}} \,$$ = E/{h∙(c/λ)} = E[W/(m2∙eV)]∙λ∙(10-9 [m])/(1.998∙10-25 [J∙s∙m/s]) = E∙λ∙5.03∙1015 [(# of photons)/(m2∙s∙eV)]

As a result of this intermediate unit conversion, the AM1.5 spectral irradiance is given in unit of the photon flux per photon energy, [number of photons/m2/s/eV], as shown in Figure 1.

Based on the above result from the intermediate unit conversion, we can derive the photon flux by numerically integrating the photon flux per photon energy with respect to photon energy. The numerically integrated photon flux is calculated using the Trapezoidal rule, as follows.

(3): $$n_{ph}(E_g) = \int_{E_g}^{infinite} \frac{dn_{ph}}{dhv} \, dhv = \sum_{i= E_g }^{infinite}(hv_{i+1} - hv_i) \frac{1}{2} (\frac{dn_{ph}}{dhv} (hv_{i+1}) + \frac{dn_{ph}}{dhv} (hv_i)) \,$$

As a result of this numerical integration, the AM1.5 spectral irradiance is given in unit of the photon flux, [number of photons/m2/s], as shown in Figure 2.

It is should be noted that there are no photon flux data in the small photon energy range from 0 eV to 0.3096 eV because the standard (AM1.5) solar energy spectrum for hν < 0.31 eV are not available. Regardless of this data unavailability, however, the graphical QE analysis can be done using the only available data with a reasonable assumption that semiconductors are opaque for photon energies greater than their bandgap energy, but transparent for photon energies less than their bandgap energy. This assumption accounts for the first intrinsic loss in the efficiency of solar cells, which is caused by the inability of single-junction solar cells to properly match the broad solar energy spectrum. However, the current graphical QE analysis still cannot reflect the second intrinsic loss in the efficiency of solar cells, radiative recombination. To take the radiative recombination into account, we need to evaluate the radiative current density, Jrad, first. According to Shockley and Queisser method,2 Jrad can be approximated as follows.

(4): $$J_{rad} = A exp(\frac{eV - E_g}{kT}) \,$$

(5): $$A = \frac{2\pi\,exp(n^2+1)E_g^2kT}{h^3c^2} \,$$

where Eg is in electron volts and n is evaluated to be 3.6, the value for GaAs. The incident absorbed thermal radiation Jth is given by Jrad with V = 0.

(6): $$J_{th} = A exp(\frac{-E_g}{kT}) \,$$

The current density delivered to the load is the difference of the current densities due to absorbed solar and thermal radiation and the current density of radiation emitted from the top surface or absorbed in the substrate. Defining Jph = enph, we have

(7): J = Jph + Jth - Jrad

The second term, Jth, is negligible compared to Jph for all semiconductors with Eg ≥ 0.3 eV, as can be shown by evaluation of the above Jth equation. Thus, we will neglect this term to simplify the following discussion. Then we can express J as follows.

(8): $$J =en_{ph} - A exp(\frac{eV - E_g}{kT}) \,$$

The open-circuit voltage is found by setting J = 0.

(9): $$eV_{OC} =E_g - kT ln(\frac{A}{en_{ph}}) \,$$

The maximum power point (Jm, Vm) is found by stetting the derivative $$\frac{dJV}{dV} \, = 0$$. The familiar result of this calculation is

(10): $$eV_{m} =eV_{OC} - kT ln(1+\frac{eV_m}{kT}) \,$$

(11): $$J_{m} = \frac{en_{ph}}{1+kT/eV_m} \,$$

Finally, the maximum work (Wm) done per absorbed photon, Wm is given by

(12): $$W_{m} = \frac{J_mV_m}{n_{ph}} \, = \frac{eV_{m}}{1+kT/eV_m} \, = eV_m - kT$$

Combining the last three equations, we have

(13): $$W_{m} = E_g - kT[ln(\frac{A}{en_{ph}}) + ln(1 + \frac{eV_{m}}{kT}) + 1] \,$$

Using the above equation, Wm (red line) is plotted in Figure 3 for different values of Eg (or nph). Now, we can fully use Henry’s graphical QE analysis, taking into account the two major intrinsic losses in the efficiency of solar cells. The two main intrinsic losses are radiative recombination, and the inability of single junction solar cells to properly match the broad solar energy spectrum. The shaded area under the red line represents the maximum work done by ideal infinite multi-junction solar cells. Hence, the limiting efficiency of ideal infinite multi-junction solar cells is evaluated to be 68.8% by comparing the shaded area defined by the red line with the total photon-flux area determined by the black line. (This is why this method is called “graphical” QE analysis.) Although this limiting efficiency value is consistent with the values published by Parrott and Vos in 1979: 64% and 68.2% respectively,3,4 there is a small gap between the estimated value in this report and literature values. This minor difference is most likely due to the different ways how to approximate the photon flux from 0 eV to 0.3096 eV. Here, we approximated the photon flux from 0 eV to 0.3096 eV as the same as the photon flux at 0.31 eV.