In the following exercise we are going to look at the density of states of two similar 2D structures:

• Graphene
• hexagonal Boron Nitride

Both feature the same 2D crystal structure with the only difference being the lattice constant and that one consists of only carbons and the other of boron and nitrogen.

Similar to the previous exercise we write the coordinates in term of the unit cell:

graphene_pdos.inp

&GLOBAL
  PROJECT graphene_pdos
  RUN_TYPE ENERGY 
  PRINT_LEVEL MEDIUM
&END GLOBAL

&FORCE_EVAL
  METHOD Quickstep
  &DFT
    BASIS_SET_FILE_NAME  BASIS_MOLOPT
    POTENTIAL_FILE_NAME  POTENTIAL

    &POISSON
      PERIODIC XYZ
    &END POISSON
    &SCF
      SCF_GUESS ATOMIC
      EPS_SCF 1.0E-6
      MAX_SCF 300

      # The following settings help with convergence:
      ADDED_MOS 100
      CHOLESKY INVERSE
      &SMEAR ON
        METHOD FERMI_DIRAC
        ELECTRONIC_TEMPERATURE [K] 300
      &END SMEAR
      &DIAGONALIZATION
        ALGORITHM STANDARD
        EPS_ADAPT 0.01
      &END DIAGONALIZATION
      &MIXING
        METHOD BROYDEN_MIXING
        ALPHA 0.2
        BETA 1.5
        NBROYDEN 8
      &END MIXING
    &END SCF
    &XC
      &XC_FUNCTIONAL PBE
      &END XC_FUNCTIONAL
    &END XC
    &PRINT
      &PDOS
        # print all projected DOS available:
        NLUMO -1
        # split the density by quantum number:
        COMPONENTS
      &END
    &END
  &END DFT

  &SUBSYS
    &CELL
      # create a hexagonal unit cell:
      ABC 2.4612 2.4612 15.0
      ALPHA_BETA_GAMMA 90. 90. 60.
      SYMMETRY HEXAGONAL
      PERIODIC XYZ
      # and replicate this cell (see text):
      MULTIPLE_UNIT_CELL 2 2 1
    &END CELL
    &TOPOLOGY
      # also replicate the topology (see text):
      MULTIPLE_UNIT_CELL 2 2 1
    &END TOPOLOGY
    &COORD
      SCALED
      C  1./3.  1./3.  0.
      C  2./3.  2./3.  0.
    &END
    &KIND C
      ELEMENT C
      BASIS_SET DZVP-MOLOPT-GTH
      POTENTIAL GTH-PBE
    &END KIND
  &END SUBSYS

&END FORCE_EVAL

The replication of the unit cell is necessary since the program samples only at he $\Gamma$ point unless instructed otherwise and we will otherwise do get a meaningful sampling of the density of states (e.g. the grid over the Brillouin Zone will be too coarse). Another option (which we will look into in the next exercise) is to sample over k-points instead.

What you will get in addition to the output file is a file named graphene_pdos-k1-1.pdos (to be precise, you will get one such file per atom kind but here we only have one, carbon) with a content similar to:

 Projected DOS for atomic kind C at iteration step i = 0, E(Fermi) =    -0.061771 a.u.
#     MO Eigenvalue [a.u.]      Occupation                 s                py                pz                px               d-2               d-1      
          d0               d+1               d+2
       1         -0.798996        2.000000        0.83986852        0.00000000        0.00000000        0.00000000        0.00000000        0.00000000        0.16013148        0.00000000        0.00000000
       2         -0.764498        2.000000        0.42353037        0.12725580        0.00000000        0.36601449        0.01397386        0.00000000        0.06436706        0.00000000        0.00485843
[...]

The columns correspond to the orbitals present in the basis set (hence projected DOS). Normally you would now do a convolution plot using a gaussian to get a smooth DOS, but instead you can also simply plot it in a scatter or vertical line plot.

Please also note the unit of the energy, it is in $E_h$. When looking at DOS plots you may want to convert it to Electronvolt instead.

While some of the new options to help with convergence are of numerical nature, the smearing is not.

• Repeat the above calculation for the different multiple cells 3x3x1, 4x4x1, 5x5x1, 6x6x1 and 7x7x1
• Even though you are not required to do any smoothing, create the DOS plots for each simulation.
• Do you see why it is necessary to do the unit cell replication? Hints: does graphene have a band gap? Compare the plots for 3x3x1 and 5x5x1.
• .. which state ($s$, $p_x$, ..) is mainly responsible for that?
• Repeat the calculations for h-BN instead (use the DZVP-MOLOPT-SR-GTH basis set for boron instead of the DZVP-MOLOPT-GTH and change the lattice constant to $2.504 Å$).
• Create the plots again (remember: now there are two pdos files, one for each kind). What is the clear difference to graphene?
• Do you now understand why we need smearing?