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Solar Energy

# Efficiency of real PV cells

In the previous chapter, we looked at the Shockley-Queisser limit, which predicts the maximum efficiency possible for an ideal solar cell. Real PV cells achieve lower performance than the SQ limit, since real PV cells don’t have ideal characteristics. This quiz will begin to explore why real performance falls below the SQ limit, and will look at how to model the performance of real PV cells.

Which of the following assumptions we used for the Shockley-Queisser limit is reasonable to use for a real cell?

In a real PV module, not all incident photons will lead to an electron hole pair. They may be reflected away by the glass cover or the electrical contacts, or they might escape the cell before being absorbed. These effects will be looked at in more detail in the following quizzes.

As a reminder, in the last chapter we found that the photon flux $$Q_p$$ from a blackbody as a function of frequency and temperature is given by: $Q_p(f,T) = \frac{2\pi f^2}{c^2}\frac{1}{e^{hf/k_BT}-1}$ The flux of photons greater than a PV cell’s bandgap $$Q_{bg}$$ can be found by integrating this spectral photon flux from $$f^*$$ to infinity, where $$f^*$$ is the frequency corresponding to the bandgap energy, given by $$f^* = E_{bg}/h$$: $Q_{bg}(f^*,T) = \int_{f^*}^{\infty} Q_p(f,T) df$ We then found the electron-hole generation rate from incident sunlight $$G_s$$ by multiplying the area of the cell by the view factor from the cell to the sun and the photon flux above the bandgap for a blackbody at a temperature $$T_{sun}$$: $G_s = AF\int_{f^*}^{\infty} Q_p(f,T_{sun}) df$

What is the correct expression for electron-hole generation rate if we introduce an absorptance (which generally varies with frequency) $$\alpha (f)$$ which is less than 100%?

In the last chapter, we found that the current-voltage relation for a PV cell is given by $I = qG_s - qR_c(V)$ Where $$q$$ is the (positive) charge of an electron, $$G_s$$ is the generation rate of electron-hole pairs due to incident sunlight, and $$R_c(V)$$ is the recombination rate of electron-hole pairs, which is generally a function of voltage. This relation holds even outside of the SQ limit, but for real PV cells $$G_s$$ and $$R_c(V)$$ will differ from the SQ limit case. What is the primary influence of reduced absorptance on electrical performance of a PV cell?

If some of these terms are unfamiliar, you can remind yourself what they are by going through the IV curve quiz.

An incident photon which doesn’t excite an electron-hole pair means there are less electrons to be collected as current. This has a direct effect on the current delivered by the PV cell, and can also have a small effect on other performance parameters like $$V_{OC}$$.

[can include some plots or interactive features here to show how changing absorptance will influence efficiency - e.g., ISC vs. absorptance, VOC vs. absorptance, FF vs. absorptance]

In the SQ limit, it is assumed that all recombination is radiative, and the recombination rate achieved is the minimum allowed by the detailed balance principle. In real PV cells, it is typical for most of the recombination to be non-radiative. In silicon, the majority of recombination is Shockley-Reed-Hall (SRH) recombination, where an excited electron drops to an intermediate level before returning to the valence band.

When all recombination was radiative, we found that recombination scaled as $$e^{V/V_c}$$, where the “thermal cell voltage” is given by $$V_c = k_BT_{cell}/q$$. For a real silicon PV cell, where most recombination is SRH recombination, will this change the scaling of recombination rate as a function of cell voltage?

While a PV cell having both radiative and SRH recombination does not change the scaling of recombination rate with voltage, it can still influence the PV cell’s performance. In the SQ limit, we found the recombination rate at zero voltage $$R_{c0}$$ by applying the detailed balance principle. Each radiative recombination event emits a photon, and the photons emitted by recombination must be matched by the incident photons from a surrounding blackbody cavity if the cell is at equilibrium.

While the detailed balance principle can help us calculate the radiative recombination rate, it does not help us calculate non-radiative recombination. Non-radiative recombination doesn’t emit a photon (at least, not one with energy equal to the bandgap), so it does not need to balance electron-hole generation from incident radiation. Non-radiative recombination will instead balance non-radiative generation of electron-hole pairs, which occurs due to thermal energy in the cell rather than incident radiation.

Suppose some of the recombination in a cell is non-radiative. How will this change the base recombination term $$R_{c0}$$ compared to the case where all recombination is radiative?

Remember that the current voltage relationship is given by: $I = qG_s - qR_c(V)$ Where the recombination rate $$R_c(V) = R_{c0}e^{V/V_c}$$. What is the primary influence of increasing the portion of non-radiative recombination on electrical performance of a PV cell?

More recombination will lead to worse electrical performance primarily through reducing $$V_{OC}$$, since the non-linearity of recombination as a function of voltage leads to recombination being most significant at high voltages.

[plots or interactives showing how recombination affects performance. e.g., ISC vs. recomb, VOC vs. recomb, FF vs. recomb]

We previously used a simple circuit model for a PV cell, which only consists of a current source connected to a diode, and a load can be connected in parallel with the diode.

Comparing to the current-voltage relation we’ve been using ($$I = qG_s - qR_c(V)$$), the current source corresponds to the photogeneration term, and flow through the diode corresponds to the recombination term (the only difference is there is no “-1” for recombination term).

Unfortunately, real PV cells aren’t electrically ideal, and include some resistances where we would like electrons to flow freely. For example, if electrical resistivity of the n-type region of a PV cell is high, there is resistance to collection of current through the load. Where would this corresponding resistance go on our PV circuit model?

Another problem is that if there are defects in the PV material, it opens a new pathway for electrons to cross the PN junction (the diode in our circuit diagram) when there’s a voltage across it. All real PV cells have some defects, which provides this alternate path for current. Where would a resistor corresponding to this alternate path go on our PV circuit model?

The electrical resistance in the n-type region of the PV cell is called series resistance, since it’s in series with the load. Series resistance must be small for good PV cell performance, otherwise power dissipates in the n-type layer rather than the load when current flows through load. The electrical resistance corresponding to the alternate path across the junction is called the shunt resistance, since it is in parallel with the load (shunt is another word for parallel in the context of electronics). Shunt resistance should be large for good PV cell performance, otherwise current flows through the shunt resistance rather than the load when there is voltage across the load.

Without these parasitic resistances, the current-voltage relation for a PV cell is given by $$I = qG_s - qR_c(V)$$. If we make a few substitutions ($$qG_s = I_{PG}$$ and $$qR_{c0} = I_0$$) this gives us a form more familiar to us for the circuit model: $I = I_{PG} - I_0 e^{V/V_c}$ What is the current voltage relation when the effects of parasitic resistances are included?

You may want to refer to the wiki page on simple circuits to help answer this question.

A large series resistance and small shunt resistance will both negatively affect PV cell performance, primarily through reduction of the fill factor, but they still influence the IV curve of a PV cell in different ways.

The series resistance acts as a voltage divider between the voltage across the load $$V$$ and the highest voltage in the “PV circuit” (the one we labeled $$V_2$$ in the solution to the previous problem. Thus, the higher the current flowing through the load, the more voltage is reduced. $$V_{OC}$$ is unaffected, since it corresponds to no current flowing, but as $$I$$ increases, $$V$$ decreases compared to the ideal case. Thus, series resistance has the overall effect of pushing the IV curve to the left, as can be seen in the figure below.

The shunt resistance offers an alternate path for current to travel without going through the load whenever there is a voltage across the shunt resistance. The higher the voltage across $$R_{shunt}$$, the more current flows through $$R_{shunt}$$ and $$I$$ is reduced. At $$I_{SC}$$, where voltage across $$R_{shunt}$$ is small (that voltage will be zero if there is no series resistance, but could start to become significant if there is a large series resistance), there is a negligible reduction in current flow through the load. As the voltage across $$R_{shunt}$$ increases, $$I$$ decreases. Thus, shunt resistance has the overall effect of pushing the IV curve down.

[[codex-parasiticresispy]]

[It would be nice to include an interactive feature here where you can increase Rseries and decrease Rshunt to see how it influences the IV curve. Like this, for example: http://www.pveducation.org/pvcdrom/solar-cell-operation/impact-of-both-series-and-shunt-resistance I’d want ours to also include FF and VOC as outputs]

Real PV cells have worse performance than the Shockley-Queisser limit. Even so, finding the SQ limit was important because it shows us the maximum performance we could hope to achieve, and because we can still use modified versions of the same equations to predict the performance of real cells (in most cases).

The code below can calculate the performance of a PV cell as a function of different parameters we’ve explored in this quiz. The default settings will provide the same results as in the SQ limit, but you can also reduce absorptance, increase recombination, and add parasitic resistances to see how performance is affected. [tool to look at performance as it varies with abs, recomb, and resistances?]

[[codex-reducedsqlimitalles]]

The next few quizzes will look at techniques to achieve good absorptance, so that as many incident photons generate electron-hole pairs as possible, and a high $$G_s$$ is maintained.

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