Shockley–Queisser Photon Utilization Limit

Semiconductor

Bandgap E_g 1.10 eV

Sun surface temperature 5778 K

Photon flux (sim speed) 1.0×

Live Readouts

Simulated conversion efficiency

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Theoretical ultimate limit (this E_g, T)

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Avg. thermalization loss / absorbed photon

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photon in flight

absorbed → useful energy E_g

heat (E − E_g thermalized)

transmitted, below E_g — lost

How it works

Sunlight arrives as a spectrum of photons whose energies follow the Planck blackbody distribution of the Sun's ~5778 K surface. A semiconductor with bandgap E_g can only turn each absorbed photon into at most E_g of usable electrical energy — any energy above E_g is thermalized as heat within picoseconds, and any photon with energy below E_g passes straight through, absorbing nothing.

photon number flux:  φ(E) ∝ E² / (exp(E / kT) − 1)     [Planck's law, kT in eV]
useful energy:        u(E) = Eg  if E ≥ Eg,  else 0
thermalized heat:     h(E) = E − Eg  if E ≥ Eg,  else 0

ultimate efficiency(Eg,T) = Eg·∫[Eg,∞] φ(E) dE   /   ∫[0,∞] E·φ(E) dE

This is the "ultimate photovoltaic efficiency" first derived by Shockley in 1961 — a simplified predecessor of the full Shockley–Queisser detailed-balance limit, which additionally subtracts unavoidable radiative-recombination losses (Voc penalty) and tops out at 33.7% for a single junction at 1.34 eV. The simpler spectral-utilization limit modeled live here peaks near 44% at E_g ≈ 1.1 eV for the Sun's spectrum — silicon's actual bandgap (1.12 eV) sits almost exactly at that spectral sweet spot.

Bandgap slider — raise it and fewer photons carry enough energy to be absorbed (more grey "transmitted" streaks); lower it and thermalization losses grow (more red heat bursts) even though almost every photon is absorbed.
Sun temperature slider — shifts the whole spectrum hotter/cooler, moving where the optimal bandgap sits.
Snap to optimal E_g — numerically sweeps E_g at the current temperature and jumps the slider to the value that maximizes the theoretical limit.
The simulated efficiency (particle counting, live) and theoretical limit (closed-form integral) should converge to the same number as counters accumulate — a direct demonstration that the statistical spectral-limit formula is exactly what the particle-by-particle bookkeeping adds up to.

absorbed 0 · thermalized 0 · transmitted 0

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