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My question is as follows, An LED, suppose say, red LED, has a single band gap, which may change subject to change of some parameters but, when I was measuring the intensities of the light of red LED, passed through a diffraction grating using a spectrometer in my experiment; say I was working with first order maxima, why I found a gaussian intensity profile centred about the diffraction angle corresponding to the wavelength that corresponds to the band gap? Why the led is emitting a finite bandwith light centred at red wavelength even though it has "only one band gap"?

Can someone explain me this?

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Why the led is emitting a finite bandwith light centred at red wavelength even though it has "only one band gap"?

IMHO, it is not quite clear what exactly the OP finds suprising here:

  • that the emission has finite bandwidth rather than a narrow emission line (like the spectrum of atoms)

OR

  • that the emission line is finite rather than infinitely broad (as we expect from a semiconductor absorption edge.)

Absorption edge
We generally expect the absorption spectrum of a semiconductor to be infinitely broad, but bounded from below by the frequency corresponding to the band gap (image source):
enter image description here

The reason for that is that (unlike for a descrete atomic spectrum) there are energy levels available for any frequency greater than that of the band gap, $\omega_g=E_g/\hbar$, with initial states in the valence band occupied and the final states in the conduction band empty.

The absorption edge increases smoothly from zero, as the density-of-state increases from the band gap edge.

In practice, there is always some absorption below the band gap:

  • discrete peaks due to the bound excitonic states
  • continuous "tails" due to the impurity levels or subbands

Diode
Light-emitting diode (just like any diode) is based not on a bulk semiconductor, but on a p-n junction: enter image description here

The emission is due to the electrons brought above the conduction band edge one n-side of the junction, transitioning to the hole states on the p-side of the junction. Thus, the range of the emitted frequencies is limited, on the one hand, by the smallest size of the band gap within the junction (which is rather close to the band gap in the bulk material), and on the other hand by the availability of the electron and hole states. This results in a finite linewidth, although it is not necessarily Gaussian (various random effects - mostly crystal imperfections - may make it however rather close to a Gaussian.)

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    $\begingroup$ Small nit to pick - both electrons and holes flow into the junction and recombine there to produce the light. I know you know that, but many folks new to semiconductors don't want to consider electrons and holes on the same footing, leading to pain later on. $\endgroup$
    – Jon Custer
    Commented Aug 16 at 13:08
  • $\begingroup$ @JonCuster indeed, I admit that I wanted the answer to remain accessible, without delving into additional explanations about electrons and holes. $\endgroup$
    – Roger V.
    Commented Aug 16 at 13:57

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