he bandgap combination of the lattice-matched design is not optimally adjusted to the solar spectrum and failed to predict such high efficiency [30] .

A rather modified analysis for prediction of such multijunction solar cells that considers the previously attained conclusions will be tried in this study. So, the potential rises through the considered multijunction solar cell will be estimated as the sum of the potential rises in the involved subcells by applying Equation (2) for each subcell. Then, it is possible to consider the multijunction as a thermopile formed of three junctions and to apply a similar relation as that applied on thermopiles to find the electromotive force or potential “ ” [27] [28] :

. (3)

In Equation (3), the flowing radiation gains a potential rise when crossing each PV junction by the Seebeck coeffi- cient of the corresponding junction times the same thermal potential of the flowing radiation “”, as the incident radiation on the three junctions belongs to the same source temperature “” and the three junctions have the same temperature “”. According to the mentioned measured data “” of each subcell, the Seebeck coefficient of these subcells can be calculated according to Equation (2) as follows:, , and. Accordingly, the total potential rise can be calculated according to (2) as follows:

(4)

Such value is identical to the measured “” of the whole multijunction cell [29] [30] . However, such result is expected according to the course of calculations, but Equation (3) may be useful for prediction of the performance of different multijunction solar cells according to the selections of grouping such photocells.

6. Thermodynamic Analysis of Efficiencies of Photovoltaic Cells

Regarding the previous conclusions of sections 2, 3 and 4, it is possible to consider the photovoltaic cell as a thermoelectrical generator driven by Seebeck effect. Hence, the efficiency of a photovoltaic cell can be expressed as a thermoelectric generator as follows [31] :

(5)

According to literature, the charges transport in conductors is characterized by energy and entropy transport [1] . So, the magnitude of output electric power can be expressed as the product of the electric potential times the rate of the flowing entropy, “”, similar to the definition of heat flow as the product of thermal potential times the rate of entropy flow, “” [3] . Introducing such expressions into Equation (5), the efficiency of the cell can be expressed as follows:

(6)

In (6), represents a scale constant for conversion the temperature from the Kelvin scale into volts [32] . The difference between the thermal entropy flow “” and the electric entropy flow “” can be determined in terms of irreversible entropy production as a measure of the possible irreversibilities in such process. According to the previous analysis and the definition of the thermoelectric effect; the conversion of thermal energy into electric energy by replacing the thermal potential by electric potential is a reversible process. So, it is possible to assume the absence of any irreversibility and to insert the following equality of the flow of electric and thermal entropy flows [33] :

. (7)

Substituting the electric potential in “E” in Equation (6) in terms of the temperature difference “” according to Equation (2) and considering the equality in Equation (7); the efficiency of the cell can be expressed as follows:

(8)

So, the cell’s efficiency may attain the Carnot cycle efficiency by increasing the Seebeck coefficient of the junction. Accordingly, the introduced multijunction technology that accumulates the resultant Seebeck coefficient of many subcells, as found in Equation (3), may lead to reach the Carnot cycle efficiency if the resultant Seebeck coefficient reaches the value of the conversion constant which is theoretically possible [34] . However, comparing such approach to the detailed balance model as introduced by Shockley and Queisser [35] , their approach had the same reasoning that considers the photovoltaic effect is influenced by the thermal potentials of the incident radiation and the photocell, but their analysis considers the source of irreversibility belongs to interactions between the incident light, as waves, and the output electric current, as traditionally considered, as electrons. In the present study, such irreversibility is absent as the incident energy and the flowing current belong to the same nature, as waves, but each have a potential that may be mutually replaced. So, the process can be considered as reversible and the efficiency may equal to the efficiency of a reversible engine operating between the assigned source and sink temperatures [34] .

7. Conclusion

Starting from a previously postulated definition of electric current as a flow of electromagnetic waves that have a specified, positive or negative, potential, it was possible to prove that the classical definition of thermoelectric effect as conversion of thermal energy into electrical energy implies the truth of such postulated definition of electric current. So, it was possible also to prove that the photoelectric and the photovoltaic effects are driven by a similar effect as the Seebeck effect which depends mainly on the thermal potential of the incident radiation and the interacting materials. We use such conclusion to deal with the photovoltaic cells as a thermoelectric power generator and to find a new limit of the efficiency of advanced and the multijunction solar cells that exceed the broken limit of Shockley and Queisser. The found limit proves the conversion of the incident thermal radiation into electric current by the photovoltaic effect is a reversible process whose efficiency equals the efficiency of a Carnot cycle operating between the temperatures of the source of radiation and of the photovoltaic junctions.

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