Energy Bands and Brillouin Zones

[Master J]

I've been studying electronic band structure in the NFE model, but first the free electron bands. I'm just a bit curious as to the exact interpretation of energy vs. wavevector plots.

The free electron plot is parabolic. I know all physically distinct solutions lie in the 1st Brillouin Zone, but in the reduced scheme, one translates higher zones into the first.

So then, one has more than one value of energy for a given k value. Does that mean that the electron, for a given k value, can have either of these energies say? These are the (quantized) energy levels?

But if the only distinct solutions lie in the 1st BZ, then howcome higher zones, give rise to higher energies?? Should all energies not be decribes in the first BZ??


I hope you can see my confusion. Regardless, any help is always greatly appreciated!

Thanks!

 

[kanato]

So then, one has more than one value of energy for a given k value. Does that mean that the electron, for a given k value, can have either of these energies say? These are the (quantized) energy levels?

Yes, that's correct.

But if the only distinct solutions lie in the 1st BZ, then howcome higher zones, give rise to higher energies?? Should all energies not be decribes in the first BZ??

Different zone schemes are equivalent, as a consequence of the periodic translational symmetry of the lattice. But usually the reduced zone scheme is the most useful. The extended zone scheme is used in this context as a trick for connecting the periodic lattice to the free electron problem, and showing how free electron bands get "folded" back to the first BZ. Outside of the connection with the free electron problem, it's not really correct or useful to think of higher zones giving rise to higher energies.

 

[Master J]

Thanks for the help.

But...I'm still not sure I get the full picture. I still don't get why ALL electron energies are not described in the first BZ. It is necessary to extend the diagram to higher zones and fold them back in, in order that we got more bands. These bands represent higher electron energies. But, just using the first BZ, where are these energies described??

 

[kanato]

All bands *are* described in the first Brillouin zone.

In a system with discrete periodicity, the pseudomomentum k is a good quantum number, and is only good up to a reciprocal lattice vector (or +/- 1/2 a reciprocal lattice vector). Beyond that, the energy bands are periodic, so $$\varepsilon_{k} = \varepsilon_{k+G}$$. In a real solid, you can only unambiguously get things in the reduced zone scheme or the periodic zone scheme. An unfolding procedure to go to the extended zone scheme is arbitrary. The free electron problem is special, because of its continuous translation symmetry. There, momentum is a good quantum number, so you can unambiguously define what you mean by the energy bands for any possible momentum.

 

[zhanghe]

yes, kanato is very correct, especially we should pay attention to these words:
"The extended zone scheme is used in this context as A TRICK for connecting the periodic lattice to the free electron problem,"