Silicon and silicon dioxide are the two that cannot make an LED — and each fails for a different reason. The answer is 3 & 4, option 3.

Why silicon fails — the direct against indirect distinction. In a direct-gap material such as GaAs the conduction-band minimum lies at the same crystal momentum as the valence-band maximum, so an electron can drop across the gap and emit a photon while conserving both energy and momentum. In an indirect-gap material such as silicon the two extrema are at different momenta, so recombination additionally requires a phonon to carry off the momentum difference. Needing three particles to meet at once makes the process improbable; recombination in silicon proceeds instead through non-radiative traps, releasing the energy as heat. Silicon is an excellent photodetector and a hopeless emitter.

Why SiO₂ fails more fundamentally. It is an insulator with a band gap of about 9 eV. No current can flow through it in the first place, so there is nothing to recombine; and a 9 eV transition would in any case lie far in the ultraviolet, well outside the visible. Its role in electronics is precisely the opposite — as the gate oxide and the isolation layer, valued for not conducting.

Why GaP is on the list despite being indirect. Pure GaP is an indirect-gap semiconductor and would be a poor emitter, but doping it with nitrogen or zinc-oxygen pairs introduces isoelectronic traps that localise an exciton. Localising the carrier in space spreads it out in momentum — a direct consequence of the uncertainty principle — which relaxes the momentum-conservation requirement and makes radiative recombination efficient. This is how the early red and green LEDs were made.

The wider pattern : LED materials are drawn from the III-V compounds and their alloys — GaAs, GaP, GaAsP, AlGaAs, InGaN — because the alloy composition can be adjusted to tune the gap, and hence the emission wavelength, across the spectrum.

Hence, the materials that are not LED materials are Si and SiO₂.