The mere fact that something is layered does not necessarily produce a defined bandgap, but e.g. if you make thin enough layers of metal, you may actually reach quantized behaviour in one dimension and still metallic behaviour in the other two dimensions.

Those materials used in photonics are simply (or not so simply?) those that happen to provide controllable properties and a suitable stability.

I think the most intuitive description starts from a Hydrogen molecule.

First something about allowed states. Picture a string held fixed at both ends (Nodes). There are a finite number of ways the string can vibrate. The lowest energy, lowest frequency longest wavelength vibration has a node at both ends and a single antinode in the middle. The next highest energy state has a third node in the middle and two antinodes with half the wavelength and twice the frequency.

A particle bound in a potential well has similar allowed states. Say you had a one dimensional hydrogen atom. The electron is attracted to the proton and so it takes energy to move it away from the proton. It’s trapped in a potential energy well. The lowest energy thing the electron can do is sit in the middle of the well right on top of the proton. However quantum mechanics won’t allow that. The electron is described as a wave function and the shape of that wave function, where the amplitude is low and where the amplitude is high, is WHERE the electron is. Because the electron is very light its wave function is much larger and more spread out than the proton. Trapped in a potential well our 1D electron wave function is a lot like the vibrations on a string. The electron can’t leave the potential well, so the wave function has to go to zero at the ends just like the fixed ends of the rope. The lowest energy mode has an antinode in the middle. You can see that this is low energy because the amplitude is large in the middle. The electron is mostly in the middle as close as possible to the proton with the lowest possible potential energy. Just like the string the electron can’t take on any energy. The vibrations have to resonate so that there are nodes at the ends or it would continually lose energy to the end constraints until it disappeared. So the next lowest energy mode possible has a node in the middle. Here it is clear that the energy is higher because less of the electron is in the middle near the proton. It‘s mostly further away in the antinodes on either side of the middle so that is higher electrical potential energy.

In three dimensions you get something very similar. The lowest energy mode is spherically symmetric with a high probability in the middle. The next highest energy mode is a dumbbell shape with a node in the middle.

Now suppose the electron is bound to two protons in close proximity. While the electron would have lower energy being right at either one or the other proton, the large size of the wave function does not allow this. We can think in terms of the deep well that is created by both atoms acting together. Here again the lowest energy mode has the electron mostly hanging out in the middle and the next higher mode has a node in the middle. With two protons the energy is an even stronger function of the electron‘s orbital. In the fundamental mode, the electron being in the middle between the protons not only puts the electron closer to the protons, but it also screens the repulsion between the two positively charged protons. The next mode with a node in the middle doesn’t screen as well and so the energy is much higher. In the lowest mode it holds the protons together (“bonding orbital”). In the higher mode the poorly screened protons will repel and could come apart “antibonding orbital”)

Now what if we have two electrons? Electrons are fermions, so they aren’t allowed to be in the same state. If one electron is in that low energy bonding state, then the other will have to be in the higher energy anti bonding orbitals, right? No, they don’t. Electrons have an extra quantum parameter: spin. If the electrons have opposite spin they both can settle into that low energy highly screened bonding orbital.

So to recap, If you bring two hydrogen atoms together, their even and odd symmetry orbitals split and separate even further in energy due to screening and separating into bonding and antibonding orbitals. However, due to spin, both electrons can settle into the low bonding orbital.

If you imagine building up a solid material by bringing in more and more atoms, this process keeps happening. Antibonding orbitals are high energy and unpopulated and bonding orbitals are lower energy and fully populated due to spin. With more electrons they still cannot be all in exactly the same state. Even the low energy bonding orbitals must split a little into slightly different energies as do the anti bonding orbitals. Soon the individual energy states blend together into a fully populated valence band and an unpopulated conduction band. In between is the band gap.

Now, in a layered structure you can create a similar effect. Different materials (where “different“ might mean different stochiometry or different doping of otherwise similar materials) have different energies of their band edges. At the interface between the materials there will be a discontinuity in the energy of the band edges. In a thin layer the discontinuity at the two interfaces can create a potential well. An electron in the valence band in the layer would have to gain energy to climb out of the well into the valence band of the adjacent material. The electron is trapped in the well and this constrains its wave function in one dimension just like that one dimensional hydrogen atom. As far as the bulk layer material is concerned the electron could take on any of the energies of the densely packed valance band. But the potential well from the thin layer further constrains the electron’s wave function. It can only have modes that fall to zero at the interfaces. So, the allowed energies become discrete and separated. There‘s a lowest energy mode with two nodes and an antinode across the layer, three nodes and two antinodes, etc. You can tailor the width of the well with the layer thickness and you can tailor the depth of the well by the choice of materials. So you can customize a one dimensional “atom” with discrete energy levels like a hydrogen atom, but you get to set the energy spacing.

You can have a look at lumped LC circuit (delay line made of discrete LC elements). It has some cut-off frequency, the signal does not propagate above that frequency. That is perhaps simplest model ( one can also use equivalent mechanical model with springs-masses) where periodic structure creates frequency bandgap.