[Physics] When does the fermi-surface fit into the first brillouin zone

Question

Talking about the free electron approximation the Fermi-surface is a circle. However, sometimes it fits into the first BZ and sometimes it does not. I read that, in general, it does not fit into the first BZ if the metal has 2 or more valence electrons.

Why is this?

Answer 1

You can understand this in the following oversimplified model. Let us consider periodic 1D chain of $N$ atoms with lattice constant $a$. Brillouin zone boundaries are located then at $\pm n\pi/a$. Let us consider three configurations:

1. Atoms have 1 valence electron on s-type orbital. In this case the energy level of this atomic orbital is split into $N$-level band. Only $N/2$ levels are occupied due to spin degeneracy. Fermi momentum is equal to $k_F = \pm\pi/2a$, hence Fermi surface is fully inside the 1st Brillouin zone.
2. Atoms have 2 valence electrons on s-type orbital. Now all $N$ levels are occupied and Fermi momentum $k_F = \pm\pi/a$, so that Fermi surface touches the edges of the 1st BZ.
3. Now let us switch to atoms with p-type orbitals occupied. If the orbitals are oriented like shown in figure below, then two energy bands of different width would emerge: one is with low mass (from two orbitals touching each other like $\infty - \infty$) and one is with large mass (from two orbitals oriented like $8 - 8$), each having $N$ levels. They have to be occupied up to the same energy level. Now, if there are 3 valence electrons, then the high mass band will be fully occupied by 2 valence electrons, whereas the third electron will be distributed among the low mass band in the 1st BZ and high mass in the 2nd BZ, as shown schematically in the figure below.

The general idea is that 2 electrons would fill one band completely and others can stay either in the first, or in the second BZ. It does not mean, that 3 valence electrons must lead to the Fermi surface protracting to the 2nd BZ. In fact, if you rotate p-orbitals in this model 45 degrees, then you will end up with the single band, where each level is degenerate not only in spin, but also in angular momentum. In this case all 3 electrons will stay in the 1st BZ.

Real band structures are quite complicated due to many factors involved, and for complex materials it often occurs that Fermi surfaces touch the BZ boundaries.