The quantum leap: photons, LEDs, solar

The red power light on your charger and the blue glow of a phone screen are not just different colors - they are photons of different energy, and that one idea is why LEDs light up, why solar panels make electricity, and why a dim blue lamp can do something a blazing red one cannot. Light, it turns out, arrives in countable lumps.

The experiment that forced light into lumps

If light were purely a smooth wave, brightness would be everything: a brighter wave carries more energy, so pile on enough of it and any electron should eventually be shaken free, regardless of color. That is not what happens. Below a certain frequency (a certain color), no electrons come off no matter how bright the light or how long you wait. Above it, they come off instantly even when the light is dim. This is the photoelectric effect, and it makes no sense for a wave. It makes perfect sense if light is a stream of particles.

The resolution, from Einstein in 1905, is that light comes in indivisible packets called photon, and one electron absorbs one photon at a time. Whether an electron escapes is a one-on-one deal: a single photon either carries enough energy to free it or it does not. Making the light brighter just sends more photons, more one-on-one deals. If each individual lump is too weak, a billion of them are still each too weak. Play with the photoelectric mode: below threshold, drag brightness to the maximum and the electron count stays stuck at zero.

One equation: energy is frequency

The energy of a single photon depends only on its frequency $f$, through a fixed constant of nature $h$ (Planck's constant):

This is stated (it is the founding postulate of quantum theory, confirmed by exactly the photoelectric measurements above), but notice how completely it explains the puzzle. Red light has a low frequency, so each red photon is a small lump of energy - below the metal's escape cost, no matter how many arrive. Ultraviolet has a high frequency, so each photon is a big lump, over the threshold on its own. Frequency sets the energy per lump; brightness only sets how many lumps. That per-photon frequency rising with energy is the same trend flagged back in electromagnetic waves and light, now made exact.

Band gaps: the energy staircase inside a material

Inside a solid, electrons cannot have just any energy. They sit in a low valence band and can only jump to a higher conduction band (where they are free to move and carry current) if something gives them a specific minimum energy. That minimum is the band gap. It is a one-jump toll: less than the gap and nothing happens, exactly the gap or more and the electron crosses. LEDs and solar cells are the same staircase run in opposite directions.

In the widget's solar mode, a photon hits the material. If its energy clears the gap, it kicks an electron up into the conduction band and makes current - that is a solar cell turning light into electricity. If the photon's energy is below the gap, the material is transparent to it and no power is harvested (which is why a solar panel wastes the low-energy part of sunlight). In LED mode the process reverses: an electron drops down across the gap and releases its lost energy as a single photon. Since $E=hf$, the gap energy fixes the frequency, which fixes the color. Slide the band gap and watch the emitted color march from red (small gap) toward blue (large gap). An LED's color is its band gap.

H = 4.1357e-15   # Planck's constant in electron-volt seconds

def photon_energy_eV(freq_hz):
    return H * freq_hz

def led_wavelength_nm(gap_eV):
    # a photon of energy = band gap; wavelength(nm) = 1240 / energy(eV)
    return 1240 / gap_eV

print(round(led_wavelength_nm(1.9)))   # -> 653 nm  a red LED
print(round(led_wavelength_nm(2.6)))   # -> 477 nm  a blue LED

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