Confused by band theory in a weak potential

[AntiElephant]

Apologies if this is rather trivial. I'm having a hard time wrapping my mind around the mathematics of energy bands and periodic potentials.

I understand that an electron in a periodic potential will be of the form $\phi_k(r) = e^{ik.r}u_k(r)$. This wavefunction has periodicity of the reciprocal lattice and so the same is true for any physical observable of the electron, such as its energy. However my confusion comes in with the introduction of energy bands.

My book begins off with the introduction of a weak periodic potential to illustrate the formation of band gaps at brillouin zone boundaries (perp. bisectors of a certain reciprocal lattice vector), and how far away from these boundaries there are two energy solutions corresponding to the free electron energies of wavevectors $k$ and $k + G$, G being the reciprocal lattice vector for the previous zone boundary.

This summarises the findings, Vg being low than the free-electron terms (E^0 denotes free-electron energy) corresponds to not being near the zone boundary.

http://img405.imageshack.us/img405/1456/40946286.png

But how does this any of this tie in with periodicity of the energy in k-space? What does it mean to have two energy eigenvalues for the same k-state? Can I have two electrons occupying the exact same state but in different energy levels? And if far away from a zone boundary of the recip. lattice vector G there correspond to two solutions, then surely there would actually correspond an infinite number of solutions for a specific k because you would be far away from the zone boundary for every recip. lattice vector G.

This;

presents a formulation of the equation above, but it doesn't have periodicity and there doesn't seem to correspond two "solutions" (or infinite if what I said before was correct) to a particular k-value as the equations before showed. I'm honestly confused with what the equations above are trying to explain in relation to band-theory.

How can we talk of periodicity in the energy if the weak periodic potential gives a form of electron energy as shown that diagram above? What is the physical significance of two energy solutions for a particular k-value?

 

[Cthugha]

This;

presents a formulation of the equation above, but it doesn't have periodicity and there doesn't seem to correspond two "solutions" (or infinite if what I said before was correct) to a particular k-value as the equations before showed. I'm honestly confused with what the equations above are trying to explain in relation to band-theory.

This image shows the extended zone scheme of the band diagram. It contains already all the info you need, but it does not show redundant info. The periodicity gives you such redundant info. In fact, periodicity will enforce that you get shifted copies of the dispersion you see in your picture. These copies will be shifted by 2 pi/a each. If you take the effort and actually draw a few (like 5 or so) next to each other, you will see that the resulting band structure is periodic. I did not find a good image of that at the moment, but have a look at the upper right panel of the following figure. This is for a more complicated system, but shows the principle.

You see that the whole set of dispersions is periodic and you get many energy states per k. However, if you know how it works, you do not need to draw all of these dispersions, but all the information is already contained in one of them. That is why people usually just show the first Brillouin zone or the extended zone scheme.

Can I have two electrons occupying the exact same state but in different energy levels?

Well, if they are in different energy levels, they are obviously not occupying the same state.

 

[AntiElephant]

This is for a more complicated system, but shows the principle.


You see that the whole set of dispersions is periodic and you get many energy states per k. However, if you know how it works, you do not need to draw all of these dispersions, but all the information is already contained in one of them. That is why people usually just show the first Brillouin zone or the extended zone scheme.



Well, if they are in different energy levels, they are obviously not occupying the same state.

Thanks for the reply. So in a non-weak periodic potential, an idential result would still hold (many bands per k-state?).

If that's all, then I have one more thing which has confused me.

The whole band system is periodic in the reciprocal lattice vectors. State k = 0 in the first band corresponds to the lowest energy possible (E = 0). However k = 0 + G = G, where G is a reciprocal lattice vector, can also exist in the first band corresponding again to the lowest energy possible. There are infinite amount of reciprocal lattice vectors. So I must have a lapse in my thought process when I question why all electrons in the periodic potential don't just occupy a k = G state in the first band and have all their energies identical to zero. They'd be in a different k-state, so Pauli would not be violated. This is obviously wrong though considering it defys everything that is sacred in Solid State Physics, but why?

 

[DrDu]

I understand that an electron in a periodic potential will be of the form $\phi_k(r) = e^{ik.r}u_k(r)$. This wavefunction has periodicity of the reciprocal lattice and so the same is true for any physical observable of the electron, such as its energy.

Just wanted to point out that the wavefunction is not periodic as long as k doesn't equal some K. only u_k is periodic.

In the weak potential picture, the two states with different energy at some K vector are sine and cosine functions, the maxima of it's square falling either into the minima or the maxima of the potential. Hence the energy difference.