Opening of a band gap - Diffraction at the edge of BZ zones?

[McLaren Rulez]

I have a hazy idea that comes from a few video lectures (see below for example) where the following is stated:

The condition for constructive interference (Bragg condition) in reciprocal space is that the wave vector should touch a Brillouin zone boundary. In this case, the standing wave that is created opens a band gap. Is this a correct way of understanding opening of band gaps and if yes, can someone explain or point me to a reference that shows why this constructive interference condition results in a band gap? I'm actually looking for intuition more than a proof to be honest.

The other way I've seen it done is using perturbation theory where the math is fine but there is no intuition on why a band gap occurs only near the edge of the Brillouin zone.

Here is a lecture that explains the connection between Bragg diffraction and the opening of a band gap but I still can't see why.

 

[eljose79]

Hello there,

Thank you for sharing your thoughts on the connection between Bragg diffraction and the opening of a band gap. I can confirm that your understanding is correct. The condition for constructive interference in reciprocal space is indeed related to the opening of a band gap.

To understand this, let's first define what a band gap is. In solid-state physics, a band gap is an energy range in a material where no electron states can exist. This means that electrons cannot occupy energy levels within this range, which results in a gap in the energy spectrum. The size of the band gap determines the material's electrical conductivity and other properties.

Now, let's look at the concept of Bragg diffraction. This phenomenon occurs when a wave (such as light or electrons) is scattered by a periodic structure, resulting in constructive interference at specific angles. In other words, Bragg diffraction happens when the wave's wavelength is in resonance with the periodicity of the structure, resulting in a strong reflection.

In reciprocal space, the periodic structure is represented by the Brillouin zone boundaries. The wave vector, which describes the wave's direction and magnitude, must touch a Brillouin zone boundary for constructive interference to occur. This means that the wave's wavelength is in resonance with the periodicity of the crystal structure, resulting in a standing wave.

Now, when a standing wave is formed, it creates a potential energy barrier for the electrons. This barrier prevents the electrons from occupying energy levels within the band gap, thus resulting in the opening of a band gap. In other words, the constructive interference condition in reciprocal space creates a periodic potential that leads to the formation of a band gap.

I hope this explanation provides you with the intuition you were looking for. If you are interested in a more detailed explanation with mathematical proofs, I would recommend referring to a solid-state physics textbook or conducting further research on the topic. Thank you for bringing up this interesting topic, and I hope this helps in your understanding.