Why Do Free Electron Energy Levels Appear as Multiple Parabolas?

[nista]

In many textbooks and sites of solid state physics energy levels for the free electron approximation in band calculations are displayed. The result is a collection of many parabolas, each of these parabolas being centered on a site of the reciprocal lattice.
This is not anyway the picture that should emerge from the free electron solution where we should find a single parabola with vertex at k=0.
Can anyone help me explain this discrepancy?
Thanks a lot

 

[kanato]

I'm not quite sure what you're asking, but it sounds like you're comparing the extended zone scheme to the free electron theory? The extended zone scheme (or any zone scheme) assumes that there is a fixed periodicity in your real space lattice, ie. the potential V(x+a) = V(x) with a > 0. This gives a periodicity in reciprocal space, in 1D the periodicity is g = 2\pi/a. This gives us a periodicity in the energy bands, \varepsilon(k) = \varepsilon(k+g). But the truly free electron model doesn't have just a finite periodicity, it has an infinitesimal periodicity, so V(x+a) = V(x) for any arbitrarily small a. If you put this into reciprocal space by taking the limit as a -> 0, you would find that g \rightarrow \infty, destroying the periodicity in reciprocal space.

So the thing to remember is that with the free electron model applied to a lattice, we assume the potential has some fixed finite periodicity, but we neglect the actual effect of that potential on the electron energy bands. This is done as a starting point for learning about reciprocal space, and it also provides something to compare real band structures to in order to see how free-electron like they are. For instance, compare the band structures on page 6 of http://www.mse.ncsu.edu/WideBandgaps/classes/MSE%20704/Handouts/BZs&Bands.pdf" .

 

[nista]

Ok that was exactly what I was asking.. I think this point is quite subtle.. thanks a lot for the explanation