What Is Smearing in Crystallography?

[PHY-101]

what exactly is a "smearing"

Hi everyone,

I would like to know what exactly is a "smearing" and a "smearing width" and its relationship with the k-point and the brillouin zone.

Could someone help me with that?

Thanks

 

[Modey3]

Smearing is a technique used to integrate the Brillouin Zone (BZ) when determining the total energy of your system. The best way to illustrate smoothing is to use sodium as an example. So let's say you decided to use a two-atom basis with simple cubic lattice vectors to generate your BCC lattice. Therefore, there are 2, 3s valence electrons per supercell. Your reciprocal lattice is therefore simple cubic and the maximum occupancy at each k-point within the BZ is 4-electrons - as opposed to 2 electrons if the 1-atom basis (just 1, 3s electron per supercell in this case) were used.

So let's say you solved the Hamiltonian and have your set of eigenvalues. Remember, for the given Hamiltonian matrix the number of eigenvalues is usually much larger than actual number of bands - especially for plane-wave basis sets. After solving the Hamiltonian eqn. you need to occupy the energy bands with electrons at each k-point. Let's say you use a Monkhorst Pack k-point grid to integrate your BZ. The grid is reduced down to the symmetrically inequivalent k-points. Now, if the maximum occupancy at each k-point is 4-electrons the BZ (or 3s band) is only half filled. This is because there are 2-electrons per supercell, an infinite number of supercells, which gives an infinite number of k-points within the BZ. To determine the Fermi Level, the BZ is occupied at each irreducible k-point scaled by a weighting factor which takes into account the symmetry of your k-point mesh. Also mind Hunds Rule is obeyed. The occupancy at each k-point is given by (# of electrons) / 4. The total occupancy of a band is given by the summation of the occupancies over all the k-points. The maximum occupancy of a band is given by the summation of the maximum occupancy ( 1 ) at each k-point. The percentage of band filling is given by the total occupancy divided by maximum occupancies. Band filling is stopped when this quantity is 1/2. Since you are using a finite number of k-points, the Fermi Level is going to depend on the spacing of your k-point grid, but will converge to the exact Fermi Level when an infinite number of k-points are used. As the k-point density increases the Fermi Level will oscillate around its exact value until the exact Fermi Level is reached at an infinite number of k-points. Partial k-point occupancies can be used to reduce this Fermi level oscillation and hence the oscillation of the total electronic energy. Partial occupancies are achieved by smearing the occupancies about the calculated Fermi Level. So instead of having occupancies of 1 or 0 at each k-point, now the occupancies near the Fermi Level can be between 0 and 1. Smearing should be used for metal due to their partially filled bands (for insulators, the Fermi Level is already known due to filled bands) I know this was long, but their isn't a short answer to this question.

modey3