# Band gaps

Beer-monster
Not sure if this should have gone in help but here goes.

I'm doing some background reading into the subject covered by my final year project for my degree. In particular I'm trying to get a grip on the topic of band gaps (acoustic to be exact but I'm looking generally at the moment).

Although I understand what a band gap is in basic terms (a gap in allowed frequency or energy levels), I've found it very hard to find a simple explanation to help me understand the cause of this effect. I'm not as good as I should be at decyphering the mysterious physics language of Jargon .

I know the effect is caused by a regular array of wave scatterers but how this produces the effect still eludes me. Is it simple interference or something more complex and how would it produce such a pronounced gap?

I hope that was clear at that someone might be able to explain something complicated to a dunce like me I have a feeling my project hinges on it

Thanks

Staff Emeritus
Gold Member
Think of the bands as being formed by the overlap of atomic orbitals.

In an isolated atom, you have electrons possesing specific discrete values of energy {given approximately by the Bohr relation E(n) = E(1)/n^2 }. It is, however more accurate to note that electron energies (according to better models) are functions of n,l,m (with no more than 2 electrons per energy level, according to Pauli). In any case, what I'm getting to is the density of states (DOS) for an isolated atom. If you plot the number of electrons (on the y-axis) vs. energy (on the x-axis) you get sharp (discrete spectrum) spikes at certain energies and nothing in between - so there are already forbidden energies for an isolated atom.

Now when you start combining several atoms, you cause these sharp spikes to spread out into a band, which is made up of many closely spaced levels. In some materials these bands (are close enough and wide enough that they) overlap and in some materials, they don't. Very simply, this is all there is to a bandgap.

Beer-monster
Thanks Goku, that clears the electric band gaps up. However how does this help overlap with wave phenomena such as light in photonic crystals.

As a stab in the dark (hope no one gets hurt)

Does the interference pattern caused when the wave scatters off an object mean that the light can only be in specific frequerncies (as the others cancel out) and then if you have many waves scattering of a regular arrangement these forbidden levels over lap like in a solid and you get a frequency band gap?

Or am I way off?

Thanks again

Photonic band gaps are almost a direct analogy to electronic band gaps. Photonic band gaps can arise when there is a periodic variation in the refractive index of a material (Analogous to a periodic variation in electronic potential in the electronic case).

The simplest case of a photonic bandgap is in the case of a high reflecting dielectric stack, or a fibre Bragg grating. These structures have a periodic modulation in the dielectric constant (which, you should recall is the refractive index squared in optical media), which result in a particular wavelength (sometimes called the design wavelength) being strongly reflected due to the reflections from all the interfaces constructively interfering.

Note however that such a bandgap only applies to light traveling normal to the interface. Such bandgaps are referred to as 1 dimensional bandgaps. There are other, more complex structures that posess 2 and even 3 dimensional bandgaps (which are termed complete bandgaps).

An example of a structure posessing a 2D photonic bandgap would be a Photonic crystal fibre (PCF). PCF's guide light, not using total internal reflection, but by exploiting resonances as was done in the 1D case. Complete photonic bandgaps are difficult to acheive, but they have been demonstrated in artificial structures and in natural strucures such as Opal.

There is much promise in using Photonic bandgap materials for use in minaturised waveguide devices, and as such is a very active field of research (My group specialises in this type of research as a matter of fact). Googling any of the above topics should yield a more comprehensive overview of this fascinating subject.

Back to your original post, bandgaps arise as a direct result of resonances between two interfaces, resonances which are in turn a direct result of interference. A strong resonance enhances transmission between two interfaces, whereas a strong antiresonance prohibits transmission. By confining light within antiresonant structures, it can be trapped and guided in the same way total internal reflection can trap light within an optic fibre.

Claude.