I want to draw the energy ($E$) diagrams for a simple cubic cell of parameter $a$, where each atom provides two electrons for the almost free electron levels for planes [100], [110] and [111].

I calculate $k$ for the first Brillouin zone:

$k_{\text{ first Brillouin zone}}=\frac{\pi}{a}=\frac{3,14}{a}$

I calculate $k_{\text{Fermi}}$:


$k_{F}=(3\pi^2n)^{1/3}=\big(3\pi^2\frac{2}{a^3}\big)=\frac{(6\pi^2)^{1/3}}{a}\approx \frac{3,90}{a}$

I calculate $k$ for the plane [100]:


$k_{[100]}=\frac{\pi}{a}=\frac{3,14}{a} = k_{\text{ first Brillouin zone}}$

I calculate $k$ for the plane [110]:



I calculate $k$ for the plane [111]:



Finally I made the drawings of the energy bands for each plane:

enter image description here

I understand why in the first zone of Brillouin there is a discontinuity as in the case of plane [100], where the material is conductor, but I do not understand for the planes [110] and [111].

  • As for planes [110] and [111], $k$ is greater than $k$ of the first Brillouin zone, should there also be in the drawing a discontinuity in the first Brillouin zone?

  • Why is it just a single semi-filled band?

  • Are they insulators or conductors in this case (planes [110] and [111])?

  • 1
    $\begingroup$ sorry that I cannot answer to your question. In any case it is not clear to me how you calculated the energy gap, that is the energy at which the second band for [100] starts. Can you kindly explain? $\endgroup$
    – gryphys
    Jan 3, 2021 at 14:54