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In physics, the Shockley–Queisser limit or detailed balance limit refers to the maximum theoretical efficiency of a solar cell using a p-n junction to collect power from the cell. It was first calculated by William Shockley and Hans Queisser at Shockley Semiconductor in 1961. The limit is one of the most fundamental to solar energy production, and is considered to be one of the most important contributions in the field.
The limit places maximum solar conversion efficiency around 30% assuming a p-n junction band gap of 1.1 eV (typical for silicon). That is, of all the power contained in sunlight falling on a silicon solar cell (about 1000 W/m²), only 30% of that could ever be turned into electricity (300 W/m²). Modern commercial single-crystalline solar cells produce about 22% conversion efficiency, the difference due largely to practical concerns like reflection off the front surface and light blockage from the thin wires on its surface.
The Shockley–Queisser limit only applies to cells with a single p-n junction; cells with multiple layers can outperform this limit. In the extreme, with an infinite number of layers, the corresponding limit is 68%.
In a traditional solid-state semiconductor, a solar cell is made from two doped crystals, one with a slight negative bias (n-type semiconductor), which has extra free electrons, and the other with a slight positive bias (p-type semiconductor), which is lacking free electrons. When placed in contact, some of the electrons in the n-type portion will flow into the p-type to "fill in" the missing electrons, also known as an electron hole. Eventually enough will flow across the boundary to equalize the Fermi levels of the two materials. The result is a region at the interface, the p-n junction, where charge carriers are depleted and/or accumulated on each side of the interface. In silicon, this transfer of electrons produces a potential barrier of about 0.6V to 0.7V.
When placed in the sun, photons in the sunlight can strike the bound electrons in the p-type side of the semiconductor, giving them more energy, a process known technically as photoexcitation. In silicon, sunlight can provide enough energy to push an electron out of the lower-energy valence band into the higher-energy conduction band. As the name implies, electrons in the conduction band are free to move about the silicon. When a load is placed across the cell as a whole, these electrons will flow out of the p-type side into the n-type side, lose energy while moving through the external circuit, and then back into the p-type material where they can once again re-combine with the valence-band hole they left behind, producing a lower-energy photon. In this way, sunlight creates an electrical current.
The Shockley–Queisser limit is calculated by examining the amount of electrical energy that is extracted per photon of incoming sunlight. There are two primary considerations.
Since the act of moving an electron from the valence band to the conduction band requires energy, only photons with more than that amount of energy will produce power. In silicon the conduction band is about 1.1 eV away from the valence band, which corresponds to red light. In other words, photons of red, yellow and blue light will all contribute to power production, whereas infrared, microwaves and radio waves will not. This places an immediate limit on the amount of energy that can be extracted from the sun. Of the 1,000 W/m² in AM1.5 sunlight, about half of that has less than 1.1 eV of energy, and will not produce power in a silicon cell. That means there is a theoretical conversion efficiency of about 50% or less, ignoring all other factors.
Another important contributor to losses is that any energy above and beyond the bandgap energy is lost; while blue light has roughly twice the energy of red light, there is no practical way to capture that energy. The electron is ejected with higher energy when struck by a blue photon, but it loses this extra energy as it travels toward the p-n junction, this energy being turned into heat in the crystal. The Shockley–Queisser limit calculation is based on this analysis, considering the difference in energy between the photons being absorbed from the sun at 6000° K and their re-emittance at room temperature at 300° K.
Another consideration is the rate of production, which does not effect the efficiency of the cell under normal conditions, but introduces further limits under certain conditions. Every photoelectron leaves behind a "photohole", an ionized atom that will attempt to catch any passing electron. If a cell were placed in conditions where a single photoelectron/hole were produced, that electron has almost no chance of re-combining with the hole it left behind - the force towards the p-n junction overwhelms the force pulling it back towards its own hole.
As the rate of photo-production increases, the number of holes in the bulk crystal grows. These are the holes left behind by previous excitations that have not yet been filled by electrons being returned to the cell. Eventually the number of holes will grow so great that every new photoelectron will meet a hole before reaching the junction, placing a limit on the rate of production. With silicon, this limiting rate is reached quite quickly, at less than "two suns" of incident light. If twice as much light is sent onto such a cell, the production rate is only slightly higher than with one sun, so the ratio of input energy to output is lower, representing a much lower efficiency. For this reason, it is not economically feasible to use mirrors or lenses to increase the production from a simple cell.
Shockley and Queisser's work considered the most basic physics only, there are a number of other factors that further reduce the theoretical power. Many of these have been explored since the 1980s. Landsberg and Baruch added various practical considerations like re-emission, while a number of researchers have attempted to characterize other losses in the cell, like interstitial defects. This latter effect explains why polysilicon cells are always outperformed by their single-crystal cousins.
Exceeding the Limit
It is important to note that the limit makes several fundamental assumptions; that the cell contains a single p-n junction, that the junction is tuned to visible light, and that any extra energy in the photons is lost. None of these assumptions is necessarily true, and a number of different attacks have been made to significantly surpass the basic limit.
The most widely explored path to higher efficiency solar cells has been to use multiple p-n junctions, each one tuned to a particular frequency of the spectrum. Since light will only react strongly with structures roughly the same size as their wavelength, as long as these layers are extremely thin they are almost transparent to lower frequencies. This allows the layers to be stacked, with the layers capturing higher frequencies (shortest wavelengths) on top, and the lower frequency light traveling through them to the lower layers.
The calculation of the fundamental efficiency limits of these "tandem cells" (or "multi-junction cells") works in a fashion similar to those for single-junction cells, with the caveat that some of the light will be converted to other frequencies and re-emitted within the structure. Using methods similar to the original Shockley-Queisser analysis with these considerations in mind produces similar results; a two-layer cell can reach 42% efficiency, three-layer cells 49%, and a theoretical infinity-layer cell 68%.
The majority of tandem cells that have been produced to date use three such layers, tuned to blue (on top), yellow (middle) and red (bottom). These cells require the use of semiconductors that can be tuned to specific frequencies, which has led to most of them being made of gallium arsenide (GaAs) compounds, often germanium for red, GaAs for yellow, and GaInP2 for blue. They are very expensive to produce, using techniques similar to microprocessor construction but with "chip" sizes on the scale of several centimeters. In cases where outright performance is the only consideration, these cells have become common; they are widely used in satellite applications for instance, where the power-to-weight ratio overwhelms practically every other cost.
Gallium arsenide has higher electron mobility than silicon, which means the photoelectrons reach their p-n junctions more quickly. It also has many more charge carriers available, which means the ratio of electrons/holes to neutral atoms is lower. These effects reduce the chance that electrons and holes will meet during the journey to the junction, which allows more light to fall on the cell before they reach equilibrium. These cells have increasing efficiency under concentrated light; under the best possible conditions and perfect lighting, a two-layer cell can reach 55% efficiency, 63% for three-layer cells, and 86% for infinite layers.
Using concentrations on the order of 500 to 1000, meaning that a 1 cm² square cell can use the light concentrated by a 31.6 x 31.6 cm Fresnel lens (covering 0.1 m² area), produces the highest efficiencies seen to date. Three-layer cells are fundamentally limited to 63%, but existing commercial prototypes have already demonstrated over 40%. These cells capture about 2/3rds of their theoretical maximum performance, so assuming the same is true for a non-concentrated version of the same design, one might expect a three-layer cell of 30% efficiency under normal sunlight. This is not enough of an advantage over traditional silicon designs to make up for their extra production costs. For this reason, almost all tandem cell research for terrestrial use is dedicated to concentrator systems, normally using mirrors or fresnel lenses.
Using a concentrator also has the added benefit that the number of cells needed to cover a given amount of ground area is greatly reduced. A conventional system covering 1 m² would require 625 cells of 16cm², but for a concentrator system only a single cell is needed, along with a concentrator. The argument for concentrated tandem cells has been that the high cost of the cells themselves would be more than offset by the reduction in total number of cells and the much lower cost of the concentrators. The downside of the concentrator approach is that at high concentrations even small movements of the sun will cause the focussed sunlight to fall off the cell, so they need to be mounted in a machine that tracks the sun as it moves. Sun-tracking system are expensive, rising with the precision required, offsetting other advantages.
To date, no large-scale tandem cell commercial systems have been deployed, although one has been planned for Spain. PV generator deployments using conventional cells are currently reaching about $5 per peak Watt for deployment and installation costs, a number the concentrator systems cannot yet match. However, Boeing's Spectrolab division claims to be aiming for $3 a Watt in the short term.
There has been some work on the use of deliberate impurities to produce mid-energy states within single crystal structures. These cells would combine some of the advantages of the multi-junction cell with the simplicity of existing silicon designs. A detailed limit calculation for these cells with a wide variety of impurities suggests a maximum efficiency of 77.2% To date, no commercial cell using this technique has been produced.
Approximately half of the solar energy reaching the Earth's surface is in the near and far infrared (IR). In silicon the energy of the bandgap is higher than the energy of these photons, and they do not contribute to energy production. Losing this energy limits cell efficiency to about 50% even if one ignores the other factors included in the Shockley–Queisser limit.
Various solutions to this problem have been proposed. The most obvious solution is to use a semiconductor with a lower bandgap that is suitable for capturing IR energy. This solution actually lowers efficiency, however, because it means more of the energy in the higher-frequency photons is lost. For this reason almost all IR-capture efforts are based on using two-layer cells with a conventional cell on top and an IR-sensitive one on the bottom. These cells have a fundamental limit the same as any other two-layer cell, at about 42%. Unlike the existing tandem cells, however, a conventional silicon cell can be used as the upper layer, which should be much less expensive to produce.
Hot electron capture
Since much of the Shockley–Queisser limit is due to energy losses between the photon energy and the energy captured from the electrons they produce, it should be no surprise that there has been a considerable amount of research into ways to capture the energy of the electrons before they can lose it in the crystal structure. A related concept is to use photoproducers that release more than one electron, instead of a single electron of greater energy. There has been a considerable amount of effort investigating quantum dots for both of these roles.
Another possibility for increased efficiency is to convert the frequency of light down towards the bandgap energy with a fluorescent material. Some fluorescent materials will convert a single high-energy photon into several lower-energy ones, although this conversion process tends to be relatively inefficient. On the upside, such a material could be painted on the front surface of an otherwise standard cell, boosting its efficiency for little cost. Overall operation of such a cell is similar to the quantum-dot case, releasing more electrons of lower energy and producing more energy overall.
A study at Ohio State University has discovered a new class of materials that can be tuned to produce electrons of any energy from light across the entire solar spectrum. In theory, these materials could capture all of the energy, and would be limited by optical issues (reflection off the front face, etc.), not the Shockley–Queisser limit.
Even without these sorts of materials, another use of fluorescence is to produce a low-cost concentrator system. In this concept, sheets of clear plastic are dyed with fluorescent paint. When the dye re-radiates the light falling on the front of the plate, it is trapped within the plastic and travels fiber optic-like to the edges of the sheet. Cells on the edges will see about 40 times concentration, far from the area of peak efficiency of GaAs cells, but without any need for tracking - light falling on the plate from any angle will still be sent to the edges.
Thermophotovoltaic cells are similar to phosphorescent systems, but use a plate to act as the downconvertor. Solar energy falling on the plate, typically black-painted metal, is re-emitted as lower-energy IR, which can then be captured in an IR cell. This relies on a practical IR cell being available, but the theoretical conversion efficiency can be calculated. For a converter with a bandgap of 0.92 eV, efficiency is limited to 54% with a single-junction cell, and 85% for concentrated light shining on ideal components with no optical losses and only radiative recombination.
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