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I have a question that surged when I was studying examples of the Fermi Surface (FS).

Now, it is my understanding that both the FS and the band diagram are defined in the Brillouin Zone (BZ) and the FS separates the occupied from the unoccupied states in the BZ, which are below and above the Fermi level in the band diagram, respectively. Therefore, this means that when drawing the band diagram, if the trajectory you are following in the BZ passes through the FS, then the band diagram will have a Fermi-level crossing at that point in the trajectory.

I tested this idea with Ca and Cu Fermi surfaces and band diagrams obtained from Materials project (https://materialsproject.org/) and the Fermi surfaces database (https://phys.ufl.edu/fermisurface/) and it seemed to work. The problem came when I looked at the alkali metals. Particularly, since the FS for all alkali metals (except Cs) are closed sphere-like surfaces, the valence band should be above the Fermi level at all the high-symmetry points except Γ (the center of the "sphere"). Furthermore, all trajectories connecting to Γ should pass through the FS and therefore be a corresponding Fermi-level crossing in the band diagram. And the valence band should be above the Fermi Level on the rest of the high-symmetry points. This is correct with points H and P, but not with N. As you can see on the diagrams in this article (https://aip.scitation.org/doi/pdf/10.1063/1.5032498) and the materials project legacy for Li (https://legacy.materialsproject.org/materials/mp-135/) as well as all the other alkali metals, the valence band is below the Fermi level at the N point, and does not even cross the Fermi level when going from Γ-N, even though the FS is be closed. This behavior only makes sense in Cs, where the FS goes into the 2nd BZ, but I just cannot make sense of it with the rest of the alkali metals. Hope it is clear. All help is appreciated.

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I think this is simply a shortcoming of the particular density functional calculations you are referencing. Your first reference is using density functional results as input for a transport calculation where these details probably aren't so important. The second explicitly warns about errors in the density functional calculation band gaps.

Here is a reference where the density functional calculations give the N point of Li above the Fermi surface. See figure 2:

https://iopscience.iop.org/article/10.1088/0953-8984/11/26/305

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  • $\begingroup$ I also think you are correct. DFT is pretty well known to be bad at getting these things correct. The hand-fitted tight-binding calculations should give a far better approximation of the Fermi surface. $\endgroup$ May 29, 2023 at 13:00

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