Obtaining a density of states (DOS)

The calculation of a density of states (DOS) is based on the presence of a self-consistent density in the working directory. With this starting point a single modification of the Fleur input file then activates the DOS generation in the next Fleur run.

The output of a DOS calculation depends on several parameters. Most important is the range of the energy mesh on which the DOS is calculated.

The generation of a DOS relies on the chosen $\vec{k}$-point set used to sample the Brillouin zone. Since this is always finite the contributions comming from each calculated state have to be broadened in energy to avoid the calculation of an overly spikey DOS that is mainly an artifact of the Brillouin zone sampling. For this the DOS is convoluted with a Gaussian distribution.

Note that the optimal choice of the standard deviation depends on the fineness of the $\vec{k}$-point mesh and also on material properties like the dispersion of the bands in the chosen energy range. The choice of the $\sigma$ parameter therefore cannot be done automatically but is a task for the user. A too small $\sigma$ yields an overly spikey DOS while a too large $\sigma$ yields an overly smooth DOS that does not show any details. The following figure demonstrates this interrelation. It shows the total DOS for Si as well as the projections of the DOS onto the $s$ and $p$ states in one of the MT spheres. For the construction of the DOS the left plot uses the default parametrization, while the other two plots modify the $\sigma$ parameter and the $\vec{k}$-point mesh for the DOS calculation (not for the SCF run).

DOS for Si with different parameters.

It can be seen that the default sigma parameter here yields an overly smooth DOS that abstracts from the details. Not even the band gap can be estimated in this way. When reducing the $\sigma$ parameter the DOS becomes very spikey because the $\vec{k}$-point density is too low to actually resolve a DOS on such a fine energy mesh. Only a reduction of $\sigma$ together with an increase of the $\vec{k}$-point density yields a DOS that is not overly spikey but resolves the details of the electronic structure. But even in this plot the $\vec{k}$-point density is not high enough to completely overcome the spikes in the range of the unoccupied bands. Of course, it is expected that such artifacts especially show up in the range of the unoccupied bands because the dispersion for unoccupied bands is typically larger than for occupied bands.

The DOS can also be obtained by employing the so called tetrahedron method for Brillouin zone integration. In this method the irreducible wedge of the Brillouin zone (IBZ) is decomposed into small tetrahedra. The eigenvalues and all related properties are only obtained at the corners of each tetrahedron. Then the eigenvalues are linearly interpolated inside the tetrahedron.

The following figure demonstrates the effect of this method on the DOS. Both calculations use the same charge density and the same number of $\vec{k}$-points inside the IBZ.

DOS for Si with histogram and tetrahedron method.

The differences between the above used histogram method and the tetrahedron method can be seen very clearly. Already with the relatively small amout of $\vec{k}$-points the DOS obtained from the tetrahedron method shows very clear features similar to the DOS with the high $\vec{k}$-point density above. Note however that the choice of states for the interpolation within each tetrahedron is naive. It is performed between the i-th eigenstates at the contributing k points. This ignores the possibility of band crossings and a DOS generated in this way may show nonphysical gaps if combined with a too small smoothening parameter sigma.