Monocrystalline Silicon - Sunlight, Silicon, and Quantum Mechanics

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It may seem a little surprising to find  ‘ quantum  mechanics ’ mentioned  in an introductory book on photovoltaics – and possibly unnerving in view of a quotation by Richard Feynman (1918 – 1988), latterly a professor at the California Institute of Technology, who received a Nobel Prize in Physics  in 1965 for his work on quantum mechanics and famously declared: ‘ I think I can safely say that nobody understands quantum mechanics ’.

So it is clear we must tread lightly, leaving the great body of 20th century quantum theory undisturbed. Yet not entirely, for it contains precious nuggets relating to the nature of sunlight and imposes fundamental limits on the efficiency of solar cells.

Certain eminent physicists, from Isaac Newton in the 17th century to Albert Einstein in the 20th, viewed light as a stream of minute particles carrying discrete packets of energy. And we stated – without explanation – that a light quantum or photon needs a certain minimum energy, known as the bandgap, if it is to have any chance of creating an electron – hole pair in a silicon crystal lattice. It is now time to bring these ideas together with the help of a little quantum theory.

The human eye is sensitive to visible light – all the colours of the rainbow from violet to red. The corresponding range of wavelengths is about 0.4 to 0.8  μm. The complete solar spectrum, previously shown in Figure 1.4 , also contains signifi cant energy at ultraviolet (UV) and especially infrared (IR) wavelengths. A key concept of quantum theory is that the energy content of a photon is related to wavelength by a surprisingly simple equation:


Where  E is the photon energy, his Planck ’ s constant, cis the velocity of light, and  λ is the wavelength. This means that the packet of energy or quantum is about twice as large for a violet photon as for a red photon. And as Einstein proposed in 1905, quanta can only be generated or absorbed as complete units.

A second key point is that solar cells based on semiconductors are essentially quantum devices. An individual solar photon can only generate an electron – hole pair if its quantum of energy exceeds the bandgap of the semiconductor material, also known as its forbidden energy gap. This is illustrated by Figure 2.13.

You may recall that the creation of an electron – hole pair involves jolting a valence electron to produce a broken bond in the crystal lattice. The electron moves from the valence bandto the conduction band,  leaving  behind an equal, but oppositely charged hole. However the energy levels of an electron in the two bands are separated by a discrete energy gap. Moving from one band to another requires a ‘ quantum leap ’ – it is all or nothing, and intermediate levels are forbidden. Long - wavelength infrared and red photons do not generally have the necessary amount of energy. Conversely most photons towards the violet end of the spectrum have more than enough and the excess must be dissipated as heat. These fundamental considerations, taken in conjunction with the Sun’s spectral distribution, reduce the theoretical maximum effi ciency of a silicon solar cell at an insolation  of  1000 W/m2 to about 45%. The fi gure does not take account of various other loss mechanisms and practical design considerations, some of which were illustrated by Figure 2.8 . So it is not hard to appreciate why cells made in research laboratories do well to reach 30% and why current commercial, mass - produced, cells achieve less than 20%.

We can now appreciate why the size of the bandgap is a very important influence on solar cell efficiency. If the bandgap is too large many photons possess insuffi cient energy to create electron – hole pairs. But if it is too small, many have a lot of excess energy that must be dissipated as heat. It is found that efficient harvesting of the Sun ’ s energy requires bandgaps in the range 1.0 – 1.6 electron volts (eV). Silicon’s bandgap of 1.1 eV is fairly good in this respect. Certain other semiconductor materials have bandgaps closer to the middle of the range, and we will discuss them later.

Unfortunately not all photons with the necessary energy are readily absorbed. Most solar cell materials, the direct - bandgap semiconductors, act as good light absorbers within layers just a few micrometres thick. But crystalline silicon, an indirect - bandgapmaterial, is not so effective. It absorbs high - energy blue photons quite easily, close to the cell ’ s top surface, but low - energy red photons generally travel much further before absorption and may exit the cell altogether. The basic problem is that successful generation of conduction electrons in silicon requires additional quantum lattice vibrations that complicate the process, so that layers less than about 1 mm thick are not good light absorbers. Special light - trapping techniques may be used to increase the pathlength of light inside the cell and give a better chance of electron – hole generation.

To summarize, it would be helpful if every photon entering a solar cell produced an electron – hole pair and contributed to power generation, in other words if the quantum effi ciencywas 100%. But quantum theory tells us this is impossible. Photons are all - or - nothing packets of energy that can only be used in their entirety. Some are too feeble in their energy content while others are unnecessarily strong, placing fundamental limits on solar cell efficiency. Disappointing though this may seem, we should always remember that sunlight is ‘ free ’ energy, to be used or not as we wish. Photons are not wasted if untapped – at least not in the sense of an old -fashioned power station burning fossil fuel that effectively discards around 60% of its precious fuel as waste heat.

Monocrystalline Silicon - Generating Power

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We have seen solar photons at work, creating minority carriers that speed towards the solar cell’s output terminals under the magical influence of the p–n junction. But how is all this internal activity reflected in the cell’s power generation, and what voltages and currents are produced at its terminals? Figure 2.9 (a) helps answer the question with an equivalent circuit summarising the cell’s behaviour as a circuit component. It consists of a diode representing the action of the p–n junction together with a current generator representing the light - generated current I L.


In dark conditions IL is zero and the cell is quiescent. If an external voltage source is connected the cell behaves just like a semiconductor diode with the characteristic shown in part (b) of the figure . We choose to define the current I as flowing into the circuit and in the dark, it must be the same as the diode current ID. Note also that  since a diode is a passive device that dissipates power, the cell’s dark characteristic lies entirely in the first and third quadrants (I and V both positive, or both negative). But if sufficient sunlight falls on the cell to turn it into an active device delivering power to the outside world, the current I must reverse and the characteristic will shift into the fourth quadrant (Inegative, V positive) shown shaded in the figure.

This equation confirms that the diode I – V characteristic is shifted down into the fourth quadrant by an amount equal to the light - generated current IL. This is shown in Figure 2.10 (a).



Most people are unfamiliar with curves in the fourth quadrant, so for convenience the I – V characteristics of a solar cell are normally ‘flipped  over’ to the fi rst quadrant. This is equivalent to plotting V against − I. Part (b) of the figure illustrates a family of such curves for a typical crystalline silicon cell rated by the manufacturer at 2 Wp. Each curve represents a different strength of sunlight, and hence a different value of IL. You will recall that PV cells and modules are normally rated in peak watts (Wp), indicating the maximum power they can deliver under standard conditions (insolation 1000 W/m2, cell temperature 25°C, AM 1.5 solar spectrum). Therefore we should first consider how the rated power of 2 W prelates to the 1000 W/m2 I – V curve.

In general the cell’s power output equals the product of its voltage and current. No power is produced on open circuit (maximum voltage, zero current) or short circuit (maximum current, zero voltage). The full rated power is obtained by operating the cell slightly below maximum voltage and current at its maximum power point (MPP), shown as P1 against the 1000 W/m2 curve, and corresponding to about 4 A at 0.5 V, or 2 W. We can only obtain the promised output power by operating the cell at its MPP. Three other curves are shown for lower insolation values of 750, 500 and 250 W/m2; each has its own MPP (P2,  P3,  P4) indicating the maximum power available from the cell at that particular strength of sunlight.

Note that the maximum voltage produced by a silicon solar cell is about 0.6 V, considerably less than the 1.5 V of a dry battery cell. This means that it is essentially a low - voltage, high - current, device and many cells must be connected in series to provide the higher voltages required for most applications. For example the PV module previously illustrated in Figure 2.1 has 72 individual cells connected in series, giving a DC voltage of about 35 V at the MPP. Higher voltages may be obtained by connecting a number of modules in series.

The  I – V characteristics suggest another important aspect of the solar cell – it is helpful to think of it as a current sourcerather than a voltage source like a battery. A battery has a more or less fixed voltage and provides variable amounts of current; but at a given insolation level the solar cell provides a more or less fi xed current over a wide range of voltage. The maximum voltage of the cell, its open - circuit  voltage  Voc, is given by the intercept on the voltage axis and lies in the range 0.5 V – 0.6 V. It does not depend greatly on the insolation. The close relationship between the diode characteristic of the p – njunction and the I – Vcharacteristics in sunlight, illustrated in Figure 2.10(a), means that the open - circuit voltage is similar to the forward voltage of about 0.6 V at which a silicon diode starts to conduct heavily.

The maximum current from the cell, its short - circuit  current  Isc, is given by the intercept on the current axis and is proportional to the strength of the sunlight. Other things being equal it is also proportional to the cell’s surface area. It represents the full fl ow of minority carriers generated by the sunlight and successfully ‘ collected ’ after crossing the p – n junction. The above parameters are further illustrated by Figure 2.11 . The blue curve shows a typical I – Vcharacteristic at 1000 W/m2 insolation, labelled with the short - circuit current, open - circuit voltage, and maximum power point. The red curve shows how power output varies with voltage; the maximum value is Pmp = Imp × Vmp. Since the current holds up well over most of the voltage range, it follows that the cell ’ s output power is roughly proportional to voltage up to the MPP. This emphasises once again the importance of operating the cell close to the MPP if its power output potential is to be realised. A widely used measure of performance that refl ects the overall quality of the cell is its fi ll factor (FF ) given by:


 An ‘ideal’ cell in which the current held right up to the short - circuit value, then reduced suddenly to zero at the MPP, would have a fi ll factor of unity. Needless to say, practical cells do not achieve this; the I – V characteristic in the fi gure has a fi ll factor of about 70%. Equation (2.4) shows that graphically it is equal to the ratio between the areas of the small and large shaded rectangles in the figure.

So far we have not considered the effects of temperature on cell performance, but actually they are quite important, especially in the case of crystalline silicon. Many people imagine that solar cells are more efficient if operated at elevated temperatures, perhaps thinking of the type of solar - thermal panel used for water heating. But solar photovoltaic cells like to be kept cool – they do very well in strong winter sunshine in the Swiss Alps! In hot climates cell temperatures can reach 70 ° C or more and system designers often go to considerable lengths to ensure adequate ventilation of PV modules to assist cooling.

The main effect of temperature on a cell’s I – V characteristic is a reduction in open - circuit voltage, illustrated by Figure 2.12 . We have repeated the 1000 W/m2 curve for the 2 Wp cell already shown in Figure 2.10 (b) for the standard temperature of 25 ° C, and added two further curves for 0 and 50°C.  The  open - circuit  voltage  changes  by  about  0.1 V  between  these extremes, corresponding to 0.33% per°C. Note that the temperature coeffi cientis negative; in other words the voltage decreases as the temperature rises. There is a much smaller effect on the short - circuit current.


 
Generally the cell loses power at elevated temperatures, a more serious effect with crystalline silicon than most other types of solar cell. You have probably noticed one major omission from this discussion – an explanation of effi ciency. At the start of this chapter we noted that commercial crystalline silicon modules have typical effi ciencies in the range 11 – 16%, but we have not so far explained the reasons for this apparently rather disappointing performance. Returning for a moment to Figure 2.10 (b) it is not clear from our discussion why this cell, which probably receives up to about 14 Wp of incident solar energy, only manages to convert  2 W p into electrical output. Where does the rest go, and why can ’ t  the efficiency be dramatically improved by better design? This raises some fundamental issues which we tackle in the next section.