The ultimte limit, of course, would be the Carnot limit with the temperature of the sun as the hot reservoir and the cell at ambient temperature of around 300 K. This will yield an efficiency limit of about 94%. However, we may have some difficulties with running a process between these reservoirs. Let us try a more realistic estimation based on the working principle of a single bandgap semiconductor solar cell.

We will try give an estimation of the maximum power we can draw from a solar cell and divide it by the incident power. There are some quite fundamental investigations on this topic [Baruch1995, Schockley1961] but we will try to stick to illustrative examples.

Let us assume that we have an ideal cell with no optical losses. The attainable current density for a given bandgap is then obtaind by integrating the spectral distribution. Starting value is the band gap energy and the integration runs over all shorter wavelengths. If we plot the value of the integral for different starting points we get the maximum current density for a given bandgap. This is the maximum current that the cell can deliver under short circuit conditions. Lower bandgaps yield higher currents because they absorb larger parts of the spectrum. This is shown for Silicon (1.1 eV) and Germanium (0.7eV) in the figure below.

Figure 1: Flux of photons per area and time and wavelength interval (red) and maximum attainable photocurrent for a given bandgap (blue, upper scale), assuming an AM1.5 spectrum.

We will consider a less general case which is more illustrative.
Assume that the charge transport is purely by diffusion of minority
carriers. Then, the relationship between current density and voltage
yields the following approximate expression between open circuit voltage
V_{oc} and short circuit current density j_{sc}:

Here, E

For the material constants of silicon and germanium and a typical doping concentration of about 10

Note that the diffusive transport is already a very ideal assumption, even in the best cells other recombination mechanisms are present which lower the maximum open circuit voltages below the discussed limits.

Figure 2: Schematic representation of dark (blue) and light (red) current voltage characteristics of a solar cell. The different quantities like open circuit voltage V_{oc}and short circuit current density j_{sc}are shown.

The ratio between the maximum power density and the product of V

The figure below shows that a broad range of bandgaps between 1.0 and 1.8 eV are suitable for use under the AM1.5 spectrum and could yield theoretical efficiencies above 20%.

Figure 3: Maximum theoretical efficiencies under AM1.5 illumination in dependence of the bandgap energy according to the ideas discussed on this page. Different assumptions in the literature can yield higher limiting values.

[Schockley1961] W. Schockley and H. J. Queisser, J. Appl Phys. 32(3) (1961) 510-519

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