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| Fig. 1: Extra-terrestrial (AM0) spectrum (black line) compared with 5250°K blackbody spectrum. [2] (Source: Wikimedia Commons) |
The Shockley-Queisser (SQ) limit, first calculated in 1961 by William Shockley and Hans-Joachim Queisser, represents the theoretical maximum efficiency of a single-junction solar cell under standard conditions. This efficiency limit, approximately 33% for a solar cell with a bandgap of 1.34 electron volts (eV) under one-sun illumination, arises due to fundamental thermodynamic and optical constraints. [1]
Underlying this fundamental limit is a number of assumptions. [2] When those assumptions are broken, one is in turn able to achieve higher efficiencies than the SQ limit predicts. Modern photovoltaic (PV) technologies are pushing beyond this limit using tandem solar cells, inter alia. By stacking multiple layers of materials with different bandgaps, tandem cells can capture a broader spectrum of sunlight and, therefore, exceed the SQ limit.
The SQ limit is based on several assumptions about the operation of a single-junction solar cell:
Single bandgap: the cell absorbs photons with energy greater than its bandgap and converts them into electricity. Photons with lower energy pass through the cell without being absorbed.
Thermalization losses: photons with energy above the bandgap lose their excess energy as heat during thermalization. That is, one can extract at most a bandgap's worth of energy from each carrier.
Radiative recombination: some electron-hole pairs recombine radiatively, emitting photons and reducing the cell's output.
No non-radiative losses: the only process by which carriers are lost is radiative recombination.
Using these assumptions, one can use so-called detailed balance arguments in order to determine the current density, J(V), of the ideal photoconverter at a given voltage, V. One argues that the total current density is the difference between the electrons excited by the number of photons exceeding the bandgap, Jsc, and a dark current Jdark arising from spontaneous emission of the electrons which are in an excited state.
The former current can be computed by integrating the entire photon flux density, bs, from the Sun. This appropriately captures that semiconductor is transparent to any light below its bandgap, Eg. The photon flux density obtains from a standard argument in equilibrium statistical mechanics, and is given as [3]
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(1) |
where E is the photon energy, Fs is a geometrical factor accounting for the angle through which photons arrive at the photoconverter (taken as π in most derivations), h is Planck's constant, c is the speed of light, k is Boltzmann's constant, and Ts is the temperature of the Sun. As mentioned then, the short-circuit current, Jsc, is computed as
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(2) |
where the integral runs from Eg to ∞. This term is not a function of applied bias and depends purely on the solar spectrum and the bandgap of the absorber in the photoconverter.
The latter current can also be computed using Planck's radiation law as the difference between the spontaneous emission when the electrons in the system have a potential V and when they are not excited at all, as in equilibrium. [4] With the electron population at a raised chemical potential μ due to the voltage of the photoconverter, Planck's radiation law changes to
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(3) |
where the subscripts e denote the photoconverter. The dark current then comes from integrating over the possible radiated energies, namely those over the bandgap. The fact that the equilibrium spontaneous current is subtracted from the dark current is because the photoconverter is still irradiated by emission from the surroundings while under illumination from the Sun. Since the raised energy electron system has a chemical potential equal to the applied voltage V of the
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(4) |
where the integral runs from Eg to ∞.
The efficiency, η, of a solar cell is given as the ratio of the power density produced (the product of voltage V and current density J(V) at that voltage, divided by the irradiation density Ps:
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(5) |
where J(V) = Jsc - Jdark(V).
Before proceeding further, it is prudent to consider an appropriate description of the irradiation density. It can be shown that the Sun is well-modelled by a blackbody at 5000-6000°K. Indeed, Fig. 1 shows the extraterrestrial spectrum (AM0) in contrast to that of a perfect blackbody at 5250°K (reduced by a factor accounting for the distance of the Earth from the Sun) with quite good agreement overall. For photoconverters on Earth (as is often the case), the appropriate spectrum is that arriving at the Earth's surface after passing through the atmosphere. A standard in the solar cell industry is the AM1.5 spectrum, the spectrum achieved after passing through 1.5 terrestrial atmospheres (associated with an angle of inclination of the sun to the solar cell). After a conventional rounding, this spectrum has a total integrated irradiance of 1000 W m-2. [2]
Using bandgap data and solar spectrum data, one can use the sequence of equations above to develop the SQ efficiency as a function of bandgap. Plotting the photoconverter efficiency as a function of bandgap under AM1.5 conditions yields the spectrum seen in Fig. 2, with a maximum efficiency of 33.16% at a bandgap of 1.34 eV. [5] As a very important reference, silicon, presently the most important material for terrestrial solar applications, has a bandgap of 1.12 eV. Thus one sees that considerations other than efficiency (e.g. manufacturability due to experience from the microelectronics industry) play a large role in silicon's dominance in terrestrial PV.
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| Fig. 2: SQ limit for single band gap cell in a 6000°K blackbody spectrum. [2] (Source: Wikimedia Commons) |
Tandem solar cells overcome the SQ limit by stacking multiple layers of semiconductors, each with a different bandgap. Since semiconductors are transparent to radiation below their bandgap, this allows the capture of the high energy part of the solar spectrum by a wide bandgap absorber on top of a low bandgap absorber, the latter of which captures the photons which pass through. This multi-junction approach allows tandem cells to capture and convert a broader range of the solar spectrum more efficiently. Intuitively, one collects more photons than otherwise since the lower bandgap absorber captures photons which otherwise would have gone uncollected; also, one has fewer losses due to thermalization, because the higher bandgap device admits less thermalization for a given photon exciting an electron across the bandgap.
For a two-absorber system, there are two conceivable arrangements: the two absorbers can be connected in series (two-terminal arrangement), or the absorbers can operate nearly independently as solar cells, including having independent electrical contacts, with the caveat that the low-bandgap absorber does not see the high energy photons filtered by the high bandgap absorber (four-terminal arrangement).
In practice, the two-terminal setup is used because of the manufacturing difficulty associated with integrating the two absorbers optically while achieving independent electrical contact. [2] The two- terminal setup immediately induces some efficiency losses from the four-terminal case under the corresponding SQ limit. There are further practical losses associated with tandems. For example, growing one absorber on top of another introduces lattice mismatches and corresponding crystal defect states which can act as recombination centers for carriers moving through the structure. [2] Furthermore, tandem cells are more expensive to produce because of the extra processing steps necessary to produce them.
© Jonathan Sharir-Smith. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] W. Shockley and H.J. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
[2] J. A. Nelson, The Physics of Solar Cells, (Imperial College Press, 2003).
[3] F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
[4] A. de Vos, Endoreversible Thermodynamics of Solar Energy Conversion (Clarendon Press, 1992).
[5] S. Rühle, "Tabulated Values of the Shockley-Queisser Limit For Single Junction Solar Cells," J. Appl. Phys. 130, 139 (2016).
