# How to identify a semi-metal from the band structure?

Wikipedia says

A semimetal is a material with a very small overlap between the bottom of the conduction band and the top of the valence band.

How small/unique the overlap should be so that we can classify a material as a semi-metal?

A rigorous definition of a semimetal can for example be found in the mathematical literature (e.g. this paper: arXiv:2002.01990v1 [math-ph], pdf version, assumption 2.5). The basic idea is that the Fermi surface, i.e. the union of points where occupied an unoccupied bands meet, consists exactly of isolated points, the Dirac points. These points are what is called "conical", meaning that you can make a linear approximation of the bands around them, such that the upper (unoccupied) and lower (occupied) bands looks like a double cone.

So essentially the overlap should consist exactly of isolated points and the bands further clearly separate at the point in the form of a cone.

• Schematic electron occupation of allowed energy bands for an insulator, metal, semimetal, and semiconductor: Here the vertical extent of the boxes indicates the allowed energy regions and the green areas indicate the regions filled with electrons.

• In a semimetal one band is almost filled and another band is nearly empty at absolute zero.

• The electron occupancy of the semimetal is slightly different from the filling of metal in which the allowed band is partially filled.

• There are many examples of semimetal in our real world, like two-dimensional material graphene. How small/unique the overlap should be so that we can classify a material as a semi-metal?

I guess the touched two bands should share an energy window between $$[0,0.005]$$ eV. And you can identify that by first-principles energy band calculation.

PS: A pure semiconductor becomes an insulator at absolute zero. The left of the two semiconductors shown is at finite temperature, with carriers excited thermally. The other semiconductor is electron-deficient because of impurities.