Origin of the energy band gap

[aaaa202]

My book has a discussion of how the Bragg model can explain the origin of energy band gaps in solid. I have attached the relevant part of this discussion. I would very much like if someone could help me understand what it's basically trying to say.
Facts:
1) In the free electron model, the eigenstates of the hamiltonian are plane waves.
2) In the nearly free electron model we consider the free electron waves to be perturbed by a periodic potential.
But what does all this have to do with Bragg reflection? that is:
1) What are the electron waves scattering against?
2) I can see the Brillioun zone boundaries fulfill the Bragg condition so what happens to the electron waves physically at these points? And what happens for the electron waves not fulfilling the Bragg condition?
3) What does the book mean by the fact that the electron waves are standing waves at the zone boundary but not in general? How can a wave be standing at one point and moving at another?

 

[Simon Bridge]

My book has a discussion of how the Bragg model can explain the origin of energy band gaps in solid. I have attached the relevant part of this discussion. I would very much like if someone could help me understand what it's basically trying to say.
Facts:
1) In the free electron model, the eigenstates of the hamiltonian are plane waves.
2) In the nearly free electron model we consider the free electron waves to be perturbed by a periodic potential.
But what does all this have to do with Bragg reflection? that is:
1) What are the electron waves scattering against?

The walls of the periodic potential.

Since there is a reflected and an incoming wave you can get a standing wave behavior that does not extend through all space.

For the Bragg condition - consider what it is a condition on...

 

[aaaa202]

It is a condition of the wavevector. But what happens for the wavevectors that do no fulfill the Bragg condition? Shouldn't they exhibit destructive interference? After all the bragg condition is a condition for constructive interference, and the exact opposite can happen too.

 

[Simon Bridge]

A condition of the wave-vector so that the wave does what?
If the wave vector satisfies the Bragg condition - what is special about it?

 

[aaaa202]

a condition on the wavevector such that the reflected waves from succesive planes are reflected in phase. If the idea is that the electron waves scatter off the potential barriers surely there are wavevectors for which the reflected waves are out of phase too? - i.e. are extinguished.

 

[Simon Bridge]

Why is it important that the reflected waves are in phase?
What would happen if they were not?

 

[aaaa202]

Isn't the whole idea behind Bragg reflection that the reflected waves are amplified by constructive interference when the Bragg condition is fulfilled?
The condition (which is equivalent to the one stated for reciprocal lattice vectors) is:
n$\lambda$=2dsin(θ)
which is derived exactly by considering when waves reflected from adjacent planes are in phase.
I don't understand why it wouldn't be important. It is basis for the whole discussion!

 

[Simon Bridge]

Why is it important that the reflected waves are in phase?

Consider: What happens when the waves are in phase that does not happen when they are not in phase?

Notice the similarity in the equation $\small n\lambda = 2d\sin\theta$ to the ones for finding diffraction maxima/minima?
In diffraction, what happens at angles where $d\sin\theta=n\lambda$ ... what happens when it is not the case?

Back to your question: you should be able to figure what happens to waves that do not fulfill the Bragg condition.

 

[aaaa202]

When the reflected waves are in phase they interfere constructively <=> Bragg condition fulfilled
When the reflected waves are in phase they interfere destructively - which happens for some of the k not fulfilling the Bragg condition.
But why is it physically that the energy vs k-vector relation is only altered for the k-vectors fulfilling the Bragg conditions. If we accept the fact that the physics behind this is all constructive and destructive interference something should happen for all k-vectors.

 

[Simon Bridge]

When the waves have interfered destructively - what happens to them?

 

[aaaa202]

they extinguish each other. Shouldn't that have some effect on the energy-kvector relation? What is it exactly that leads us from constructive interference to the formation of energygaps? I don't understand the physics behind all of this completely.

 

[DrDu]

Let's try something different: Assume that the crystal potential V(x) is very small. Then you can try to calculate it's effect in perturbation theory.
If you start from $\psi_0=N\exp(ikr)=|k>$ and $E_k=k^2/2m$. Now $V_{kk'}<k'|V|k>$ will only be different from zero if k'=k-K where K is some reciprocal lattice vector.
Now, if $|\Delta E_{kk'}|=|E_{k'}-E_k|>>V_{kk'}$, the perturbation is unimportant. But if $|E_{k'}-E_k|\approx V_{kk'}$, we have to use degenerate perturbation theory in lowest order and obtain the perturbed energies $E_k1$ and $E_{k2}$ as solutions of the two by two Schroedinger equation $H\psi=E-(E_k+E_k')/2)\psi$ with H being the two by two matrix $\Delta E \sigma_z+V_{kk'} \sigma_x$ where $\sigma_{x,z}$ are Pauli matrices.
You will find that the splitting $2V_{kk'}$ when k' fulfills $|k'|=|k|$, i.e. if scattering is elastic. You can derive from this condition the Laue and Bragg conditions.
Try to plot this explicitly for the one-dimensional case where $K=2\pi n/a$. Try also to find the explicit form for the two eigenfunctions when the Laue condition is fulfilled.

 

[Simon Bridge]

The wavelengths depend on the energies.
Some wavelengths reinforce and some destruct - so there will be some wavelengths and thus energies present and some absent ... the absent energies make for gaps.

I always preferred watching what happens to the discrete energy levels as you add more wells together.
You can model it easily with finite square wells - for a singe finite square well, you have no trouble understanding why there are energy gaps right?

... put two together and you get two energy levels close together where you had one before ... it's like the level has doubled up. Three wells and you get three levels and so on. For very many wells, you get lots of closely spaces levels and at some point they are too close together to tell from a continuum. Those are the bands.

The gaps between bands are where the forbidden zones between discrete energy states did not get filled by all the doubling up. Not all materials have these gaps... in some, the bands touch or overlap.

 

[aaaa202]

that's a nice way to picture it, though maybe not so relevant for my question.
You say some wavelengths reinforce and some destruct. And my whole point has been that it doesn't seem that destructive interference has anything to say. Because the energy bands are formed at the zone boundaries (i.e. at the wavelengths reinforcing), while nothing happens anywhere else (especially for the wavelengths that destruct).

 

[DrDu]

that's a nice way to picture it, though maybe not so relevant for my question.
You say some wavelengths reinforce and some destruct. And my whole point has been that it doesn't seem that destructive interference has anything to say. Because the energy bands are formed at the zone boundaries (i.e. at the wavelengths reinforcing), while nothing happens anywhere else (especially for the wavelengths that destruct).

As I tried to explain, what singles out the zone boundaries is that there constructive interference can happen between states which are energetically degenerate in zeroth order but at other points only between states with different energy.

 

[Simon Bridge]

Why would anything happen anywhere else?
Maybe you have to actually do the math to get this one? Crunch some numbers?