10.7: A continuum of quantum states - quantum numbers in a crystal

In a crystal quantum states can be indexed by the Bloch quantum number $k$. In the LCAO approximation, there is a state for each possible atomic orbital at each value of ${\bf k}$. As the number of electrons tends to infinity, the allowed ${\bf k}$’s form a continuum. The most important application of quantum mechanics in solid state physics is to understand the relationship between energy and momentum. A graph of energy vs momentum is called a band structure.

States are occupied from the lowest energy upwards according to the exclusion principle. The set of momenta which correspond to the maximum allowed energy form a surface in the 3-d space - the so-called Fermi Surface.

Shown is the valence “band structure” of dhcp potassium calculated using DFT and pseudopotentials: letters are crystallographic notation for values of ${\bf k} (\Gamma = (0, 0, 0)$, others are on the edge of the Brilloiun zone). Note the free electron parabola around $\Gamma$, as $E = \hbar^2 k^2/2m$. This structure has four layers of atoms per unit cell, so on average there are two bands below the Fermi surface at each $k$-point (each is spin degenerate). There are lots of bands crossing the Fermi level, showing that electrons can move from one state to another without requiring energy: potassium is a metal. $\Gamma$-A is quite a short distance in k-space, corresponding to waves along the long direction in the unit cell: the band structure appears like a parabola “folded back” on itself.

Figure $\PageIndex{1}$: Band structure of potassium, energy scaled so that $E_F=0$. x-axis labels denote a path through the 3d space of ${\bf k}$-vectors.