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## Efficiency limits of photovoltaic energy conversion

One of the most frequently asked questions in photovoltaics is about efficiencies. Usually people are little impressed by almost 25% for single junction record cells and more than 30% in case of tandems. Surprisingly, nobody ever askes this question for the latest sportscar even though the efficiency of typical combustion engines in cars is probably not much higher. Despite its development for more than a century and the input of quite significant amounts of money. Anyway...

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.

### Current density

In the chapter on illumination we discussed the spectral irradiation density in W/nm.m2. By dividing with the corresponding photon energy it is easy to calculate the the number of photons per area and wavelength interval. For the estimation of the photocurrent we need to know about absorption processes in the semiconductor; photons with an energy less than the bandgap cannot be absorbed, photons with higher energies can excite an electron hole pair, their excess energy is lost. The figure below shows the number of photons per area and time (the flux density) in dependence of the wavelength for the AM1.5 spectrum.

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.

### Open circuit voltage

An upper limit for the open circuit voltage of a solar cell will obviously be the band gap. Lower values are due to recombination processes in the cell. Even in an otherwise ideal material, black body radiation will present a minimum amout of unavoidable radiative recomibation, simply because we operate the solar cell at some finite temperature. The assumption that all light with energy higher than the band gap energy is ideally absorbed defines the solar cell as a black body for this spectral region. As such it will also emit black body radiation because it operates at some non-zero temperature and it is in equilibrium with its environment and the solar irradiation. This concept is called detailed balance [Schockley1961].

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 Voc and short circuit current density jsc: Here, Eg is the bandgap and jsc the short circuit current. The current prefactor j00 for the diffusion current of minority carriers in a p-type absorber involves the effective densities of states NC and NV, the doping concentration NA, the minority carrier diffusion constant Dn, and the minority diffusion length Ln. For the material constants of silicon and germanium and a typical doping concentration of about 1016 cm-3 we obtain j00 in the range from 108 to 109 mA/cm2. After putting this into the above equations we find that the open circuit voltages are about 400 mV lower than the band gap energy. Indeed, the open circuit voltages for high efficiency germanium, silicon and gallium arsenide solar cells are up to 245, 706 and 1020 mV, respectively, with their respective bangaps of 0.67, 1.12 and 1.43 eV.

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.

### Fill Factor

Now we know the maximum current densitiy and the maximum voltage for a given band gap energy. However, at both points of the characteristics the actual output power of the solar cell is zero. We must find the conditions of maximum output (MPP, maximum power point). In the figure below this point corresponds to the largest rectangle which we can fit between the origin and the part of the curve in the fourth quadrant. Figure 2: Schematic representation of dark (blue) and light (red) current voltage characteristics of a solar cell. The different quantities like open circuit voltage Voc and short circuit current density jsc are shown.

The ratio between the maximum power density and the product of Voc times jsc is called fill factor (FF). We assume a value of 80% but for high efficiency GaAs cells it is as high as 87%.

### Efficiency limit

For a calculation of the resulting efficiency for a given bandgap we use the following equation: 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.

### References

[Baruch1995] P. Baruch, A. De Vos, P. T. Landsberg, and J. E. Parrot, Sol. En. Mat. 36 (1995) 210-222
[Schockley1961] W. Schockley and H. J. Queisser, J. Appl Phys. 32(3) (1961) 510-519

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