Density of state (DOS)

Introduction

• Density of States (DOS) represents the number of available electronic states at some energy.
• It is a fundamental concept in solid-state physics and materials science, providing crucial information about the electronic structure and properties of materials.

Physical meaning of DOS

Fundamental concept

• DOS represents the number of states available for electron occupation at energy $E$.
• Higher DOS indicates more states available for electron occupation, which is often useful for electronic conduction, making/breaking chemical bonds, magnetization etc.
• The energy region with (1) non-zero DOS, (2) below the Fermi energy is called the valence band, and the region with (1) non-zero DOS, (2) above the Fermi energy is called the conduction band. The zero-DOS region between the valence and conduction bands are called the band gap.

Mathematical Definition

• Formal definition: $$DOS(E) = \sum_i^k \delta(E - \varepsilon_{i,k})$$ where:
  • $\varepsilon_{i,k}$: the energy eigenvalues
  • $i$ runs over bands
  • $k$ runs over k-points in the Brillouin zone
  • $\delta$ is the Dirac delta function
• In practice, $\delta$ function is replaced by a finite energy range ($\Delta E$), so the DOS means the number of electronic state within this energy gap.

Integration Properties

• Number of electrons: $$N = \int^{E_F} DOS(E)dE$$
• Total energy: $$E = \int^{E_F} E \cdot DOS(E)dE$$
• $E_F$ is the Fermi energy.

Practical DOS Calculations

Detailed Calculation Steps

• Usually, the geometry optimization is done before the DOS calculation.
• If the LDOS tag is set to .TRUE., the DOS calculation is done at the last step of geometry optimization. So separate single-point calculation at the optimized geometry is usally not necessary.
• If you want to do DOS calculation at higher k-point mesh or higher cutoff energy, you can take the separate calculation.

INCAR

• No specifical tag is necessary to output the total DOS, but LORBIT need to be set when PDOS is needed (mentioned later).
• Note that the correct DOS cannot be obtained unless the number of k-points is sufficiently large.

• The smearing parameters in the INCAR (ISMEAR, SIGMA) also affect the accuracy of the DOS.

• Structure Optimization:

  IBRION = 2    # Ion relaxation
  NSW = 100     # Number of ionic steps
  

• DOS Calculation:

  IBRION = 0
  NSW    = 0
  LDOS   = .TRUE.  # actually this is default
  

After VASP calculation

• A file called DOSCAR will be output.
• Be aware that the output differs between spin-polarized and non-polarized calculations.
• For non-polarized calculations, the meaning of each column in DOSCAR is as follows: energy DOS integrated-DOS

Making plottable DOS file

• DOSCAR has many sections, so it is not easy to make plot from it.
• It is useful to use vaspkit for this purpose.

Installing vaspkit

1. Go to vaspkit website: https://vaspkit.com/index.html
2. Go to the "latest release page" (in SourceForge).
3. Find the latest tar.gz file for Linux, and get the URL.
4. Go to your VASP-installed-computer (supercomputer).
5. Download the tar.gz file by using wget (wget URL).
6. Extract with tar zxvf vaspkit-xxx.tar.gz.
7. cd vaspkit.x.x.x
8. source setup.sh
9. source ~/.bashrc

Using vaspkit

• vaspkit is quite easy to use. Just type vaspkit in terminal, and follow the instruction.
• Total DOS
  • To generate the DOS-related functions, type 11.
  • To get the total DOS, type 111.
  • Total DOS file TDOS.dat is generated. The energy position is shifted so as the Fermi energy becomes 0.

Plotting the DOS

• Copy *.dat file generated from vaspkit. Now it is easy to make plot.
• Any software (Excel, Gnuplot, etc) is OK.
• When using gnuplot, the command becomes like

  gnuplot
  > plot "TDOS.dat" using 1:2 with lines
  > plot "TDOS.dat" using 1:2 with lines, "TDOS.dat" using 1:3 with lines # when spin-polarized case
  

Projected DOS (PDOS)

• The PDOS refers to the DOS decomposed by angular momentum (s, p, d, etc.).
• To get the PDOS, use LORBIT tag in INCAR.
  • LORBIT = 10: makes s-, p-, d-decomposed DOSs.
  • LORBIT = 11: makes s-, px-, py-, pz-, dxy- ... DOSs.
• Plotting procedure is same with the total DOS.
• If you want to take the information of the d-band (to calculate the d-band center, for example), you should use PDOS.

Localized DOS (LDOS)

• LDOS refers to the DOS calculated separately for each atom.
• LDOS can be obtained from vaspkit.
• If you want to focus on the oxygen atom 2p band (often denoted as $\rm {O}_{2p}$ band), both localization and projection is needed.