Shockley–Queisser Efficiency Limit Calculator

Introduction to the Shockley–Queisser single-junction limit

This Shockley–Queisser limit calculator estimates the detailed-balance efficiency ceiling for a single-junction solar cell from three physical inputs: absorber band gap, source temperature, and cell temperature. It answers a deliberately idealized question: how efficiently could one junction convert light if radiative recombination were the only unavoidable loss? The result is therefore a theoretical benchmark, not a forecast for a commercial panel.

That boundary makes the calculation useful. It lets a material researcher compare likely band gaps, lets a student see the current–voltage tradeoff, and gives a designer a quick check on whether a proposed absorber is in a plausible spectral range. A real device will also face reflection, imperfect collection, contact resistance, nonradiative recombination, and weather-dependent illumination. Those effects belong in a later device model, after this upper-bound comparison has done its job.

What the Shockley–Queisser efficiency question measures

The Shockley–Queisser model treats the source as a blackbody spectrum and the solar cell as a radiatively limited diode. Photons with energy below the band gap cannot make electron–hole pairs and pass through unused. Photons above the gap can contribute to current, but only approximately the band-gap energy is available per collected carrier pair; the excess energy is lost as thermalization. Raising the gap tends to increase voltage but rejects more of the incoming spectrum. Lowering it captures more photons but reduces voltage. The famous efficiency peak comes from balancing those competing effects.

Cell temperature matters because a warmer junction emits more thermal radiation and generally loses voltage headroom. The source temperature shapes the photon distribution reaching the cell. The default 5778 K source approximates the Sun as a blackbody, while 300 K is a common reference temperature for the cell. Keeping those assumptions visible is important when comparing one run with another.

How to use the Shockley–Queisser calculator

Enter the absorber band gap in electronvolts, then enter the source and cell temperatures in kelvin. Select Compute limit to calculate the ideal short-circuit current, open-circuit voltage, fill factor, and efficiency for that set of assumptions. The form validates the intended ranges, but sensible input still depends on describing one coherent physical scenario.

1. Start with a candidate material’s band gap, E_g, in eV.
2. Use the effective blackbody temperature of the light source for T_sun.
3. Use the expected junction temperature, not a Celsius value, for T_cell.
4. Compare the current, voltage, fill factor, and efficiency together rather than using only the percentage.

For a clean comparison between materials, hold both temperatures fixed and change only the band gap. For a thermal sensitivity check, hold the band gap and source temperature fixed while changing the cell temperature. This one-variable-at-a-time approach makes the direction of each tradeoff much easier to understand.

Inputs for the solar-cell detailed-balance estimate

Band gap energy E_g (eV) is the energy threshold of the absorber. It determines the lowest-energy photon that can create a useful carrier pair in this simplified model. A value around 1.3–1.4 eV is often near the single-junction sweet spot under a Sun-like spectrum, although the precise result depends on assumptions.

Sun temperature T_sun (K) describes the ideal blackbody source spectrum. It is not the air temperature outdoors or a panel temperature. Cell temperature T_cell (K) is the junction’s operating temperature. Kelvin is required for both temperature fields: 300 K is approximately 27 °C. Do not enter Celsius directly, and do not substitute joules for eV in the band-gap field.

Formulas behind the Shockley–Queisser limit

The calculator numerically integrates the blackbody photon flux above the band gap. In compact form, the spectral photon flux used by the detailed-balance model is proportional to the following expression, where E is photon energy, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, and T is absolute temperature:

The short-circuit current is found by integrating the diluted source flux from E_g upward, while the dark radiative current is found by integrating the cell’s own emission over the same energy range. The voltage follows the diode relation, and the output power is the product of voltage, current, and fill factor. Finally, efficiency is output power divided by the incident source power:

This implementation uses numerical integration and an ideal-diode fill-factor approximation. It is designed for an instructive, fast screening estimate rather than for reproducing every optical and electrical detail of a laboratory device.

Worked example: a 1.34 eV absorber under a Sun-like source

Leave the defaults at E_g = 1.34 eV, T_sun = 5778 K, and T_cell = 300 K, then compute the limit. This is a useful baseline because the band gap is close to the range where the single-junction current–voltage compromise is favorable under a Sun-like blackbody spectrum. The results panel reports a comparatively strong balance of current collection and voltage potential under radiative-limit assumptions.

Next, lower only the band gap to 1.07 eV. More low-energy photons are now available, so short-circuit current tends to rise, but open-circuit voltage falls. Raise only the gap to 1.61 eV instead and the voltage side improves, while the loss of usable photons can reduce current substantially. The maximum efficiency is not simply the largest current or the largest voltage; it is the most favorable product after fill factor is included.

Interpreting the detailed-balance result

The short-circuit current J_sc represents the ideal current density available when the terminals are effectively shorted. Open-circuit voltage V_oc represents the maximum voltage side of the radiative diode model. Fill factor describes how closely the current–voltage curve can approach the rectangle defined by those two extrema. Efficiency η combines all three against the incoming power.

A higher current number is not automatically a better absorber, and neither is a higher voltage number. Use the efficiency as the summary measure, then look at current and voltage to understand why it moved. The Copy summary button is useful for keeping a brief record of a run when comparing several candidate gaps.

Limitations of this Shockley–Queisser solar-cell estimate

The Shockley–Queisser limit assumes one junction, radiative recombination, an idealized blackbody source, and ideal carrier collection above the band gap. It excludes nonradiative pathways, reflection and transmission losses, incomplete absorption, series and shunt resistance, spectral mismatch, angular effects, concentration, degradation, tandem architectures, and module packaging. It also does not claim that a material can actually be fabricated with ideal electronic quality.

Use the calculation as a ceiling and a comparison tool. A measured cell should normally sit below it, sometimes far below it, for understandable engineering reasons. If you need a performance forecast for a particular device, combine this result with measured optical absorption, external quantum efficiency, current–voltage curves, thermal conditions, and realistic illumination data.