Can semimetals be explained by the nearly free electron model? In the quasi-free electron model, only U shaped and flipped U shaped parabolic energy bands emerge.
So I think one can not derive from the free electron model anything about semimetals, as there need to be two U shaped bands next to each other for a material to have a band overlap and be considered a semimetal, which does not occur in this model.
In an old exam, I read tough that one can deduce from the quasi-free electron model that semimetals have small potentials, as the quasi-free electron models predict the bandgap to be proportional to the potential strength.
I wonder whether this statement makes actually sense. Because no matter how small the bandgap, there will never be a semimetal in the quasi-free model, right?
 A: While band overlap can't occur in the free electron model in one dimension it can in two- or more dimensions. In these higher dimensions there don't have to be two U shaped bands next to each other in order to have band overlap.
To convince one of this one should draw a $k_x-k_y-E(\vec{k})$ graph in the first BZ and look at it from the side. In the unperturbed case $E(\vec{k})$ is quadratic. The energy $E(\vec{k})$ on the corners of the first BZ (e.g. at $(\pi/2,\pi/2)$) is therefore higher than on the edges of the BZ between the corners, especially in the middle (e.g. at ($(\pi/2,0)$). The dispersion looks like a rag which is suspended from the four corners of the BZ.
When looking at the dispersion from the side (e.g. view on the $k_x-E$ plane) one therefore sees a U shape. The minimum of the U is in the middle of an BZ edge and the two maxima are on the two corners of that edge. The difference in energy from the middle to the corner may be called $\Delta$.
If $\Delta$ is bigger than the band gap, which is caused by the perturbing potential $V(x)$ then there will be an band overlap. Because in this case the second band in the middle of the BZ edge starts below the energy of the first band at the corners.
The band overlap is small if the perturbing potential is weak. The quasi free electron model with a weak potential may therefore indeed be used to model a semimetal.
