The Electronic Density of States (DOS)¶

Authors: Dou Du, Taylor James Baird and Giovanni Pizzi

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Source code: https://github.com/osscar-org/quantum-mechanics/blob/master/notebook/band-theory/density_of_states.ipynb

This notebook demonstrates various approaches for the numerical calculation of the density of states (DOS) for a 3D free-electron model with periodic boundary conditions.

Goals¶

• Familiarize yourself with various numerical methods employed to calculate the electronic density of states.
• Examine the resulting DOS and compare the accuracy and computational cost of the various methods.

Background theory¶

More on the background theory.

Tasks and exercises¶

1. Investigate the influence of the number of k-points on the resulting DOS.

   Solution

   In the right panel, the plotted blue line is the analytical result for the DOS of the free electron model. By choosing different numbers of k-points via the "Number of k-points slider", we can investigate how the quality of the calculated results varies with the density of the k-point mesh. You will observe that the numerical results converge to the analytical result with increasing number of k-points. This can be attributed to the fact that the DOS can be interpreted as a probability density of electronic states as a function of energy. Since energy is generally related to the k-vector magnitude, the quality with which we resolve the range of energy eigenvalues is in turn directly controlled by how fine our sampling of the k-point mesh is.

2. Which method gives most accurate results? Which method is fastest and why?

   Solution

   The linear tetrahedra interpolation (LTI) method is an accurate numerical approach which linearly interpolates the 3D k-points grid within a set of tetrahedra into which reciprocal space has been subdivided (see the background theory notebook). The LTI method can yield much better results compared to a simple histogram. Gaussian smearing renders the histogram plot considerably smoother, and closer to the analytical solution. The histogram method is a simple statistical representation of the eigenvalues (one simply represents the frequency with which a given range of eigenenergies occur). This latter approach is typically the fastest of those discussed, but shall also generally give results with the poorest level of resolution.

3. Set the number of k-points to the maximum value and consider Gaussian smearing with a reasonable value for the smearing parameter (say $\sigma=0.07$). Looking at the calculated DOS with the G-vector range set to $0$, how does it compare with the analytical result? Can you explain any discrepancies you notice? Increasing the G-vector range to $1$, how does the calculated DOS now compare?

   Solution

   You will notice that even with a high number of k-points and an appropriate smearing parameter, the calculated DOS starts to deviate significantly from the analytical result above an energy of around . This discrepancy can be attributed to the fact that when we only consider the G-vector $G=0$, we are neglecting to include several bands in the electronic energy dispersion that still contribute energy eigenvalues that fall within the range we are interested in. That is, and you can verify the claim by increasing the G-vector slider to the value 1, several bands are folded back into the first Brillouin zone. You can see which bands undergo folding by tracking those which change from black to red upon incrementing the G- vector slider.

Interactive visualization¶

(be patient, it might take a few seconds to load)

Legend¶

(How to use the interactive visualization)

Interactive figures¶

The left panel shows the electronic bandstructure of a free electron gas in 3 dimensions while the right panel displays the corresponding density of states. You can choose the number of k-points used in the computation of the DOS by manipulating the k-points slider. Similarly, the number of G-vectors employed in the calculation can be varied with the "G-vector range" slider (the value is capped at 1 to avoid excessive computational time in the case of the LTI method, however one should still expect long waiting times in in this case when $N_k$ is large).

Controls¶

Three buttons allow one to compute the DOS with the three methods discussed earlier. The calculated DOS will appear in the figure on the right, superimposed on the analytic curve. Computing the DOS with a large number of k-points may take several seconds. When using the simple histogramming method, you can adjust the number of bins employed by varying the "Number of bins" slider. For the Gaussian smearing method, you can also tune the standard deviation, $\sigma$, of the Gaussian functions by adjusting the "Gaussian $\sigma$" slider.