Shockley–Queisser limit

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The Shockley-Queisser limit for the efficiency of a solar cell. The curve is wiggly because of IR absorption bands in the atmosphere. In the original paper,[1] the solar spectrum was approximated by a smooth curve, the 6000K blackbody spectrum. As a result, the efficiency graph was smoother and the values were slightly different.

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.[1] 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.[2]

The limit places maximum solar conversion efficiency around 33.7% assuming a single p-n junction with a band gap of 1.34 eV (using an AM 1.5 solar spectrum).[1] That is, of all the power contained in sunlight falling on an ideal solar cell (about 1000 W/m²), only 33.7% of that could ever be turned into electricity (337 W/m²). The most popular solar cell material, silicon, has a less favourable band gap of 1.1 eV, resulting in a maximum efficiency of about 32%. Modern commercial mono-crystalline solar cells produce about 24% conversion efficiency, the losses 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 86% using concentrated sunlight.[3]


The Shockley-Queisser limit, zoomed in near the region of peak efficiency.

In a traditional solid-state semiconductor such as silicon, a solar cell is made from two doped crystals, one an n-type semiconductor, which has extra free electrons, and the other a p-type semiconductor, which is lacking free electrons, referred to as "holes." When initially placed in contact with each other, some of the electrons in the n-type portion will flow into the p-type to "fill in" the missing electrons. 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 0.6 V to 0.7 V.[4]

When the material is placed in the sun, photons from the sunlight can be absorbed in the p-type side of the semiconductor, causing electrons in the valence band to be promoted in energy to the conduction band. This process is known as photoexcitation. As the name implies, electrons in the conduction band are free to move about the semiconductor. When a load is placed across the cell as a whole, these electrons will flow from the p-type side into the n-type side, lose energy while moving through the external circuit, and then go back into the p-type material where they can re-combine with the valence-band holes they left behind. In this way, sunlight creates an electric current.[4] (The process is similar if the photons are absorbed in the n-type side of the semiconductor; the only difference is that instead of the photo-excited electrons flowing from the p-type side into the n-type side, the photo-excited holes flow from the n-type side into the p-type side. Both processes then involve electrons from the conduction band of the n-type side moving around the external circuit to recombine with the holes in the valence band of the p-type side.)

The limit[edit]

The Shockley–Queisser limit is calculated by examining the amount of electrical energy that is extracted per photon of incoming sunlight. There are three primary considerations:

Blackbody radiation[edit]

Any material, that is not at absolute zero (0 Kelvin), emits electromagnetic radiation that can be approximated as blackbody radiation. In the case of a solar cell at ambient room temperature, at 300 Kelvin, a baseline energy is always being emitted. This energy cannot be captured by the cell, and represents about 7% of the available incoming energy.

This radiation effect is dependent on cell temperature. Any energy lost in a cell is generally turned into heat, so any inefficiency in the cell increases the cell temperature when it is placed in sunlight. As the temperature of the cells increases, the blackbody radiation also increases, until an equilibrium is reached. In practice this equilibrium is normally reached at temperatures as high as 360 Kelvin, and cells normally operate at lower efficiencies than their room temperature rating. Module datasheets normally list this temperature dependency as TNOCT (NOCT - Nominal Operating Cell Temperature).

Radiative recombination[edit]

Black curve: The limit for open-circuit voltage in the Shockley-Queisser model (i.e., voltage at zero current). The red dotted line shows that this voltage is always below the bandgap. This voltage is limited by recombination.

Absorption of a photon creates an electron-hole pair, which could potentially contribute to the current. However, the reverse process must also be possible, according to the principle of detailed balance: an electron and a hole can meet and recombine, emitting a photon. This process reduces the efficiency of the cell. Other recombination processes may also exist (see "Other considerations" below), but this one is absolutely required.

Spectrum losses[edit]

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 a photoelectron. In silicon the conduction band is about 1.1 eV away from the valence band, which corresponds to infrared 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.[5] 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 19% of that has less than 1.1 eV of energy, and will not produce power in a silicon cell. 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, that energy is not captured by devices with a single p-n junction. 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 (the energy is converted into heat).[5] This accounts for about 33% of the incident sunlight, meaning that from spectrum losses alone there is a theoretical conversion efficiency limit of about 48%, ignoring all other factors.

All together[edit]

Considering the spectrum losses alone, a solar cell has a peak theoretical efficiency of 48%. Thus the spectrum losses represent the vast majority of lost power. Including the effects of blackbody radiation and recombination, the efficiency is described by the following equation:[6]

\eta=\frac{qV(\phi_s-\phi_r)}{\sigma T_{sun}^4}

where q is the electric charge, V is voltage across the device, \phi_s is the incident photon flux entering the device, \phi_r is the radiative photon flux leaving the device, \sigma is the Stefan–Boltzmann constant, and T_{sun} is the temperature of the sun. A single-junction cell has a theoretical peak performance of about 33.7%, or about 337 W/m² in AM1.5.[1][5]

Other considerations[edit]

Shockley and Queisser's work considered the most basic physics only, there are a number of other factors that further reduce the theoretical power.

Limited mobility[edit]

When an electron is ejected through photoexcitation, the atom it was formerly bound to is left with a net positive charge. Under normal conditions, the atom will pull off an electron from a surrounding atom in order to neutralize itself. That atom will then attempt to remove an electron from another atom, and so forth, producing an ionization chain reaction that moves through the cell. Since these can be viewed as the motion of a positive charge, it is useful to refer to them as "holes", a sort of virtual positive electron.

Like electrons, holes move around the material, and will be attracted towards a source of electrons. Normally these are provided through an electrode on the back surface of the cell. Meanwhile, the photoelectrons are moving forward towards the electrodes on the front surface. For a variety of reasons, holes in silicon move much more slowly than electrons. This means that during the finite time while the electron is moving forward towards the p-n junction, it may meet a slowly moving hole left behind by a previous photoexcitation. When this occurs, the electron recombines at that atom, and the energy is lost (normally through the emission of a photon of that energy, but there are a variety of possible processes).

Recombination places an upper limit on the rate of production; past a certain rate there are so many holes in motion that new electrons will never make it to the p-n junction. In silicon this reduces the theoretical performance under normal operating conditions by another 10% over and above the thermal losses noted above. Materials with higher electron (or hole) mobility can improve on silicon's performance; gallium arsenide (GaAs) cells gain about 5% in real-world examples due to this effect alone. In brighter light, when it is concentrated by mirrors or lenses for example, this effect is magnified. Normal silicon cells quickly saturate, while GaAs continue to improve at concentrations as high as 1500 times.

Non-radiative recombination[edit]

Recombination between electrons and holes is detrimental in a solar cell, so designers try to minimize it. However, radiative recombination—when an electron and hole recombine to create a photon that exits the cell into the air—is inevitable, because it is the time-reversed process of light absorption. Therefore, the Shockley-Queisser calculation takes radiative recombination into account; but it assumes (optimistically) that there is no other source of recombination. More realistic limits, which are lower than the Shockley–Queisser limit, can be calculated by taking into account other causes of recombination. These include recombination at defects and grain boundaries.

In crystalline silicon, even if there are no crystalline defects, there is still Auger recombination, which occurs much more often than radiative recombination. By taking this into account, the theoretical efficiency of crystalline silicon solar cells was calculated to be 29.4%.[7]

Exceeding the limit[edit]

Breakdown of the causes for the Shockley-Queisser limit. The black height is energy that can be extracted as useful electrical power (the Shockley-Queisser efficiency limit); the pink height is energy of below-bandgap photons; the green height is energy lost when hot photogenerated electrons and holes relax to the band edges; the blue height is energy lost in the tradeoff between low radiative recombination versus high operating voltage. Designs that exceed the Shockley-Queisser limit work by overcoming one or more of these three loss processes.

It is important to note that the analysis of Shockley and Queisser was based on the following assumptions:

  1. One electron–hole pair excited per incoming photon
  2. Thermal relaxation of the electron–hole pair energy in excess of the band gap
  3. Illumination with unconcentrated sunlight

None of these assumptions is necessarily true, and a number of different approaches have been used to significantly surpass the basic limit.

Tandem cells[edit]

The most widely explored path to higher efficiency solar cells has been multijunction photovoltaic cells (also called "tandem cells"). These cells use multiple p-n junctions, each one tuned to a particular frequency of the spectrum. This reduces the problem discussed above, that a material with a single given bandgap cannot absorb sunlight below the bandgap, and cannot take full advantage of sunlight far above the bandgap. In the most common design, a high-bandgap solar cell sits on top, absorbing high-energy, low-wavelength light, and transmitting the rest. Beneath it is a lower-bandgap solar cell which absorbs some of the lower-energy, longer-wavelength light. There may be yet another cell beneath that one, with as many as four layers in total.

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% in un-concentrated sunlight.[3]

The majority of tandem cells that have been produced to date use three 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 consideration. They also can be used in concentrated photovoltaic applications (see below), where a relatively small solar cell can serve a large area.

Tandem cells are not restricted to high-performance applications; they are also used to make moderate-efficiency photovoltaics out of cheap but low-efficiency materials. One example is amorphous silicon solar cells, where triple-junction tandem cells are commercially available from Uni-Solar and other companies.

Light concentration[edit]

Sunlight can be concentrated with lenses or mirrors to much higher intensity. The sunlight intensity is a parameter in the Shockley-Queisser calculation, and with more concentration, the theoretical efficiency limit increases somewhat. (If, however, the intense light heats up the cell, which often occurs in practice, the theoretical efficiency limit may go down all things considered.) In practice, the choice of whether or not to use light concentration is based primarily on other factors besides the small change in solar cell efficiency. These factors include the relative cost per area of solar cells versus focusing optics like lenses or mirrors, the cost of sunlight-tracking systems, the proportion of light successfully focused onto the solar cell, and so on.

A wide variety of optical systems can be used to concentrate sunlight, including ordinary lenses and curved mirrors, fresnel lenses, arrays of small flat mirrors, and luminescent solar concentrators.[8][9] Another proposal suggests spreading out an array of microscopic solar cells on a surface, and focusing light onto them via microlens arrays,[10] while yet another proposal suggests designing a semiconductor nanowire array in such a way that light is concentrated in the nanowires.[11]

Intermediate band photovoltaics[edit]

There has been some work on producing 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 infinite bands suggests a maximum efficiency of 77.2%[12] To date, no commercial cell using this technique has been produced.

Photon upconversion[edit]

Main article: Photon upconversion

As discussed above, photons with energy below the bandgap are wasted in ordinary single-junction solar cells. One way to reduce this waste is to use photon upconversion, i.e. incorporating into the module a molecule or material that can absorb two or more below-bandgap photons and then emit one above-bandgap photon. Another possibility is to use two-photon absorption, but this can only work at extremely high light concentration.[13]

Thermal photon upconversion[edit]

Thermal upconversion is based on the absorption of photons with low energies in the upconverter, which heats up and re-emits photons with higher energies.[14] The upconversion efficiency can be improved by controlling the optical density of states of the absorber[15] and also by tuning the angularly-selective emission characteristics. For example, a planar thermal upconverting platform can have a front surface that absorbs low-energy photons incident within a narrow angular range, and a back surface that efficiently emits only high-energy photons.[16] A hybrid thermophotovoltaic platform exploiting thermal upconversion was theoretically predicted to demonstrate maximum conversion efficiency of 73% under illumination by non-concentrated sunlight. A detailed analysis of non-ideal hybrid platforms that allows for up to 15% of absorption/re-emission losses yielded limiting efficiency value of 45% for Si PV cells.

Hot electron capture[edit]

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.[17] One system under investigation for this is quantum dots.[18]

Multiple exciton generation[edit]

A related concept is to use semiconductors that generate more than one excited electron per absorbed photon, instead of a single electron at the band edge. Quantum dots have been extensively investigated for this effect, and they have been shown to work for solar-relevant wavelengths in prototype solar cells.[18][19]

Another, more straightforward way to utilise multiple exciton generation is a process called singlet fission (or singlet exciton fission) by which a singlet exciton is converted into two triplet excitons of lower energy. This allows for higher theoretical efficiencies when coupled to a low bandgap semiconductor[20] and quantum efficiencies exceeding 100% have been reported.[21]

Fluorescent downconversion/downshifting[edit]

Another possibility for increased efficiency is to convert the frequency of light down towards the bandgap energy with a fluorescent material. In particular, to exceed the Shockley–Queisser limit, it is necessary for the fluorescent material to convert a single high-energy photon into several lower-energy ones (quantum efficiency > 1). For example, one photon with more than double the bandgap energy can become two photons above the bandgap energy. In practice, however, this conversion process tends to be relatively inefficient. If a very efficient system were found, such a material could be painted on the front surface of an otherwise standard cell, boosting its efficiency for little cost.[22] In contrast, considerable progress has been made in the exploration of fluorescent downshifting, which converts high-energy light (e. g., UV light) to low-energy light (e. g., red light) with a quantum efficiency smaller than 1. Dyes, rare-earth phosphors and quantum dots are actively investigated for fluorescent downshifting.[23] For example, silicon quantum dots enabled downshifting has led to the efficiency enhancement of the state-of-the-art silicon solar cells.[24]

Thermophotovoltaic downconversion[edit]

Main article: Thermophotovoltaic

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.[25]


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