In physics, the radiative efficiency limit (also known as the detailed balance limit, Shockley–Queisser limit, Shockley–Queisser efficiency limit or SQ limit) is the maximum theoretical efficiency of a solar cell using a single p–n junction to collect power from the cell where the only loss mechanism is radiative recombination in the solar cell. It was first calculated by William Shockley and Hans-Joachim Queisser at Shockley Semiconductor in 1961, giving a maximum efficiency of 30% at 1.1 eV. The limit is one of the most fundamental to solar energy production with photovoltaic cells, and is one of the field's most important contributions.
This first calculation used the 6000K black-body spectrum as an approximation to the solar spectrum. Subsequent calculations have used measured global solar spectra, AM 1.5, and included a back surface mirror which increases the maximum solar conversion efficiency to 33.16% for a single-junction solar cell with a bandgap of 1.34 eV. That is, of all the power contained in sunlight (about 1000 W/m2) falling on an ideal solar cell, only 33.7% of that could ever be turned into electricity (337 W/m2). The most popular solar cell material, silicon, has a less favorable 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 of the cell and light blockage from the thin wires on the cell surface.
The Shockley–Queisser limit only applies to conventional solar cells with a single p-n junction; solar cells with multiple layers can (and do) outperform this limit, and so can solar thermal and certain other solar energy systems. In the extreme limit, for a multi-junction solar cell with an infinite number of layers, the corresponding limit is 68.7% for normal sunlight, or 86.8% using concentrated sunlight (see Thermodynamic efficiency limit and solar-cell efficiency).
Background
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 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.
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.
The limit The Shockley–Queisser limit is calculated by examining the amount of electrical energy that is extracted per photon of incoming sunlight. There are several considerations:
Blackbody radiation Any material, that is not at absolute zero (0 kelvins), emits electromagnetic radiation through the black-body radiation effect. In a cell at room temperature, this represents approximately 7% of all the energy falling on the cell.
Any energy lost in a cell is turned into heat, so any inefficiency in the cell increases the cell temperature when it is placed in sunlight. As the temperature of the cell increases, the outgoing radiation and heat loss through conduction and convection also increase, until an equilibrium is reached. In practice, this equilibrium is normally reached at temperatures as high as 360 Kelvin, and consequently, cells normally operate at lower efficiencies than their room-temperature rating. Module datasheets normally list this temperature dependency as (NOCT - Nominal Operating Cell Temperature).
For a "blackbody" at normal temperatures, a very small part of this radiation (the number per unit time and per unit area given by , "c" for "cell") is photons having energy greater than the band gap (wavelength less than about 1.1 microns for silicon), and part of these photons (Shockley and Queisser use the factor tc) are generated by recombination of electrons and holes, which decreases the amount of current that could be generated otherwise. This is a very small effect, but Shockley and Queisser assume that the total rate of recombination (see below) when the voltage across the cell is zero (short circuit or no light) is proportional to the blackbody radiation . This rate of recombination plays a negative role in the efficiency. Shockley and Queisser calculate to be 1700 photons per second per square centimetre for silicon at 300K.
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.
In the Shockley–Queisser model, the recombination rate depends on the voltage across the cell but is the same whether or not there is light falling on the cell. A factor fc gives the ratio of recombination that produces radiation to total recombination, so the rate of recombination per unit area when V = 0 is 2tcQc/fc and thus depends on Qc, the flux of blackbody photons above the band-gap energy. The factor of 2 was included on the assumption that radiation emitted by the cell goes in both directions. (This is actually debatable if a reflective surface is used on the shady side.) When the voltage is non-zero, the concentrations of charge carriers (electrons and holes) change (see Shockley diode equation), and according to the authors the rate of recombination changes by a factor of exp(V/Vc), where Vc is the voltage equivalent of the temperature of the cell, or "thermal voltage", namely
(q being the charge of an electron). Thus the rate of recombination, in this model, is proportional to exp(V/Vc) times the blackbody radiation above the band-gap energy:
(This is actually an approximation, correct so long as the cell is thick enough to act as a black body, to the more accurate expression This paper finds the same open-circuit voltage and short-circuit current as de Vos and Pauwels, but does not give the correct function for .
The difference in maximum theoretical efficiency however is negligibly small, except for tiny bandgaps below 200meV.)
The rate of generation of electron-hole pairs not due to incoming sunlight stays the same, so recombination minus spontaneous generation is
where
(Shockley and Queisser take fc to be a constant, although they admit that it may itself depend on voltage.)
The rate of generation of electron-hole pairs due to sunlight is
where is the number of photons above the band-gap energy falling on the cell per unit area, and ts is the fraction of these that generate an electron-hole pair. This rate of generation is called Ish because it is the "short circuit" current (per unit area). When there is a load, then V will not be zero and we have a current equal to the rate of generation of pairs due to the sunlight minus the difference between recombination and spontaneous generation:
The open-circuit voltage is therefore given (assuming fc does not depend on voltage) by
The product of the short-circuit current Ish and the open-circuit voltage Voc Shockley and Queisser call the "nominal power". It is not actually possible to get this amount of power out of the cell, but we can get close (see "Impedance matching" below).
The ratio of the open-circuit voltage to the band-gap voltage Shockley and Queisser call V. Under open-circuit conditions, we have
Asymptotically, this gives
or
where is the voltage equivalent of the temperature of the sun. As the ratio goes to zero, the open-circuit voltage goes to the band-gap voltage, and as it goes to one, the open-circuit voltage goes to zero. This is why the efficiency falls if the cell heats up. In fact this expression represents the thermodynamic upper limit of the amount of work that can be obtained from a heat source at the temperature of the sun and a heat sink at the temperature of the cell.
Spectrum losses 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 an electron-hole pair. In silicon the conduction band is about 1.1 eV away from the valence band, this corresponds to infrared light with a wavelength of about 1.1 microns. In other words, photons of red, yellow and blue light and some near-infrared will contribute to power production, whereas radio waves, microwaves, and most infrared photons will not.C. S. Solanki and G. Beaucarne, "Advanced Solar Cell Concepts", Interuniversity Microelectronics Center, Belgium 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). This accounts for about 33% of the incident sunlight, meaning that, for silicon, from spectrum losses alone there is a theoretical conversion efficiency limit of about 48%, ignoring all other factors.
There is a trade-off in the selection of a bandgap. If the band gap is large, not as many photons create pairs, whereas if the band gap is small, the electron-hole pairs do not contain as much energy.
Shockley and Queisser call the efficiency factor associated with spectrum losses , for "ultimate efficiency function". Shockley and Queisser calculated that the best band gap for sunlight happens to be 1.1 eV, the value for silicon, and gives a of 44%. They used blackbody radiation of 6000K for sunlight, and found that the optimum band gap would then have an energy of 2.2 kTs. (At that value, 22% of the blackbody radiation energy would be below the band gap.) Using a more accurate spectrum may give a slightly different optimum. A blackbody at 6000 K puts out 7348 W per square centimetre, so a value for of 44% and a value of photons per joule (corresponding to a band gap of 1.09 V, the value used by Shockley and Queisser) gives equal to photons per second per square centimetre.
Impedance matching