The emitted wavelength is fixed by the band gap, so start there. When an electron in the conduction band recombines with a hole in the valence band, the photon carries away the band-gap energy:

\(E_{photon}=E_{g}=h\nu=\dfrac{hc}{\lambda}\)

which in practical units becomes the formula worth memorising:

\(\lambda\ (\mu\text{m})=\dfrac{1.24}{E_{g}\ (\text{eV})}\)

Apply it to gallium arsenide, whose band gap at room temperature is 1.42 eV:

\(\lambda=\dfrac{1.24}{1.42}=0.873\ \mu\text{m}=873\ \text{nm}\)

The visible spectrum ends at about 700 nm, so 873 nm lies clearly in the near infra-red — option 4.

Why GaAs is used for LEDs at all, despite being invisible: it is a direct band-gap material. An electron at the bottom of the conduction band sits directly above a hole at the top of the valence band in momentum space, so recombination needs no phonon to conserve momentum and the transition is fast and efficient. Silicon and germanium are indirect, which is why they make excellent detectors but hopeless emitters.

Where the infra-red output is exactly what is wanted. GaAs emitters drive remote controls, optocouplers, IR proximity and obstacle sensors, and the first generation of short-haul fibre links — silicon photodiodes happen to be most sensitive near 850 nm, so detector and emitter match beautifully.

To move into the visible, the band gap must be widened, which is done by alloying: adding phosphorus to give GaAsP shifts the emission through red and amber, and GaP reaches green. Longer wavelengths for long-haul fibre (1310 and 1550 nm) come from InGaAsP, whose gap is narrower still.

Hence, GaAs LEDs emit in the infra-red region.