Solar Energy
# PV: Engineering and Advanced Concepts

Recall the expression for calculating power collected by a single-junction cell (with the bandgap energy \(E^*\)) in the ultimate efficiency case: \[P_{PV} = \int_{E^*}^{\infty} B_E \frac{E^*}{E} dE\] Where \(B_E\) is the spectral irradiance of the radiation source illuminating our PV cell (i.e., the sun). For now, we’ll use Planck’s law and approximate the solar spectrum as a blackbody at \(\SI{5800}{\kelvin}\).

Note: in the past we’ve used the spectral irradiance as a function of wavelength (\(B_{\lambda}\)) and frequency (\(B_f\)), but here we are using spectral irradiance as a function of photon energy (\(B_E\)), since it is the most convenient way to directly choose the bandgap energies we want to use. \[B_E = \frac{2\pi E^3}{h^3 c^2}\frac{1}{e^{E/k_BT}-1}\]

If we have a PV cell with two bandgaps of \(E_1^*\) and \(E_2^*\), and a way of sorting photons to the correct junction, what is the expression for power collected? We will use the convention that higher subscripts correspond to higher energy, so \(E_2^* > E_1^*\)

We can find the ultimate efficiency of a PV cell by taking the electrical power collected by the cell and dividing by the total power incident on the cell, which can be calculated by integrating over all incident frequencies: \[P_{tot} = \int_0^{\infty} B_E dE\] \[\eta = \frac{P_{PV}}{P_{tot}}\]

The contour plot below shows ultimate efficiency versus different values of \(E_1^*\) and \(E_2^*\) for a two junction cell:

What is the highest ultimate efficiency (in percent) that can be achieved for a 2J cell?

The maximum ultimate efficiency in the Shockley-Queisser limit (with a single junction) is just under 44% for a bandgap of \(\SI{1.06}{\electronvolt}\). Using a 2J cell with the optimum bandgap values of \(\SI{0.77}{\electronvolt}\) and \(\SI{1.6}{\electronvolt}\), the ultimate efficiency can be increased to just above 60%.

Adding more junctions can improve efficiency further. The expression for power collected by a multi-junction cell (only considering spectral losses) follows a pattern that should be familiar if we compare the single junction case to the two junction case. For a multi-junction cell with \(N\) junctions, where the bandgap of the nth junction corresponds to the energy \(E_n^*\), the total collected power can be given by:

\[P_{PV} = \sum\limits_{n=1}^N \int_{E_n^*}^{E_{n+1}^*} B_E \frac{E_n^*}{E} dE\]

With \(E_{N+1}^* = \infty\). The code environment below uses this equation to calculate the ultimate efficiency of multi-junction cells with various bandgaps. You can change the values in the list “bgs” to change the bandgap energies. The values in this list will be used as the bandgap energies (in \(\si{\electronvolt}\)) for each junction.

Running the code will calculate the ultimate efficiency for a multi-junction cell with the specified bandgap energies. It will also output a plot with a curve showing the solar irradiance spectrum (approximated as a blackbody at \(\SI{5800}{\kelvin}\)) incident on the cell, and filled shapes showing the portion of the spectrum that is successfully collected as electricity in the ultimate efficiency case.

Does adding a new junction always increase ultimate efficiency?

Note: the code is all written for you, so you only need to change the list “bgs” if you want to explore the influence of bandgap energies on ultimate efficiency. However, if you’re interested in looking at the specifics of the calculation, the code includes commenting and documentation, so you can see exactly how the program is written.

Using multi-junction cells, we could potentially reach ultimate efficiencies much higher than what is possible with single junction cells, and in fact, some multi-junction cells have already demonstrated efficiencies higher than the SQ limit.

With infinite junctions, the ultimate efficiency could even reach 100%. However, as real single junction cells can’t achieve the ultimate efficiency value in practice, real MJ cells are also unable to reach their corresponding ultimate efficiency. The next quiz will look at real implementations of MJ cells and how to calculate more realistically achievable efficiencies for them.

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