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Graphene †

In this tutorial, the converge the Fermi level with respect to the number of k-points is discussed.

Structural optimization †

First of all, cell relaxation is performed by using "vc-relax" using the following input file

&control
 calculation  = 'vc-relax'
 restart_mode = 'from_scratch',
 pseudo_dir   =  '/home/ikutaro/QE/pseudo',
 outdir       = './tmp'
 prefix       = 'gr'
 tstress      = .true.
 tprnfor      = .true.
/
&system
 ibrav       = 4
 A           = 2.46
 C           = 10.00
 nat         = 2
 ntyp        = 1
 ecutwfc     = 40.0
 ecutrho     = 400.0
 occupations = 'smearing'
 smearing    = 'mp'
 degauss     = 0.02
/
&electrons
 diagonalization = 'david'
 conv_thr        = 1.0e-12
 mixing_beta     = 0.5
/
&ions
/
&cell
 press          = 0.0d0
 press_conv_thr = 0.5d0
 cell_dofree    = 2Dxy
/
ATOMIC_SPECIES
 C  12.01070 c_pbe_v1.2.uspp.F.UPF
ATOMIC_POSITIONS (crystal)
  C 0.0000000000000000 0.0000000000000000 0.0000000000000000
  C 0.3333333333333333 0.6666666666666667 0.0000000000000000
K_POINTS (automatic)
 12 12 01 0 0 0

In this example, GBRV ultrasoft pseudopotential, cutoff energies of 40 and 400 Ry, and nonshifted 12 x 12 uniform k-point mesh is used. The vacuum of 10 Angstrom is inserted to separate periodic images. For the cell relaxation in two dimension, 'cell_dofree=2Dxy' is used.

SCF and band structure calculations †

Having relaxed the cell parameters, let us start the band structure calculation. In this tutorial, band structure calculations are performed by varying the k-point mesh in the SCF calculation to learn how the number of k-points or k-point density is crucial to determine the position of Fermi level and thereby the location of the Dirac point at the K point.

• SCF calculation

  &control
   calculation  = 'scf'
   restart_mode = 'from_scratch',
   pseudo_dir   =  '/home/ikutaro/QE/pseudo',
   outdir       = './tmp'
   prefix       = 'gr'
   tstress      = .true.
   tprnfor      = .true.
  /
  &system
   ibrav       = 4
   A           = 2.46575870
   C           = 10.00
   nat         = 2
   ntyp        = 1
   ecutwfc     = 40.0
   ecutrho     = 400.0
   occupations = 'smearing'
   smearing    = 'mp'
   degauss     = 0.02
   nbnd        = 8
  /
  &electrons
   diagonalization = 'david'
   conv_thr        = 1.0e-12
   mixing_beta     = 0.5
  /
  &ions
  /
  &cell
   press          = 0.0d0
   press_conv_thr = 0.5d0
   cell_dofree    = 2Dxy
  /
  ATOMIC_SPECIES
   C  12.01070 c_pbe_v1.2.uspp.F.UPF
  ATOMIC_POSITIONS (crystal)
    C 0.0000000000000000 0.0000000000000000 0.0000000000000000
    C 0.3333333333333333 0.6666666666666667 0.0000000000000000
  K_POINTS (automatic)
   12 12 01 0 0 0

• Band structure

  &control
   calculation  = 'bands'
   restart_mode = 'from_scratch',
   pseudo_dir   =  '/home/ikutaro/QE/pseudo',
   outdir       = './tmp'
   prefix       = 'gr'
   tstress      = .true.
   tprnfor      = .true.
  /
  &system
   ibrav       = 4
   A           = 2.46575870
   C           = 10.00
   nat         = 2
   ntyp        = 1
   ecutwfc     = 40.0
   ecutrho     = 400.0
   occupations = 'smearing'
   smearing    = 'mp'
   degauss     = 0.02
   nbnd        = 8
  /
  &electrons
   diagonalization = 'david'
   conv_thr        = 1.0e-12
   mixing_beta     = 0.5
  /
  &ions
  /
  &cell
   press          = 0.0d0
   press_conv_thr = 0.5d0
   cell_dofree    = 2Dxy
  /
  ATOMIC_SPECIES
   C  12.01070 c_pbe_v1.2.uspp.F.UPF
  ATOMIC_POSITIONS (crystal)
    C 0.0000000000000000 0.0000000000000000 0.0000000000000000
    C 0.3333333333333333 0.6666666666666667 0.0000000000000000
  K_POINTS (crystal_b)
  4
    0.00000000  0.00000000  0.00000000 20.0
    0.66666667 -0.33333333  0.00000000 20.0
    0.50000000  0.00000000  0.00000000 20.0
    0.00000000  0.00000000  0.00000000  0.0

  In this example, the energy band is draw along the Gamma-K-M-Gamma line.
• Post-processing To plot the band structure with gnuplot or alike, "bands.x" is used. Here's an example of the input file:

  &bands
   outdir  = './tmp'
   prefix  = 'gr'
   filband = 'gr.bands'
   lsym    = .true 
  /

  With this input, "gr.bands.gnu" and "gr.bands.rap" are generated. For use with gnuplot, "gr.bands.gnu" can be used. One can use the following script ("band_plot.sh") to generate the band plot with gnuplot

  #!/bin/sh
   ef=`grep Fermi gr.scf.out | awk '{print $5}'`
   echo $ef
  gnuplot <<EOF
  ef=${ef}
  xmin=0.0
  xmax=1.5784
  ymin=-21.0
  ymax=4.0
  g1=0.0
  k=0.6671
  m=1.0007
  g2=1.5784
  offset=0.5
  set arrow from k,ymin to k,ymax nohead lt 0
  set arrow from m,ymin to m,ymax nohead lt 0
  unset xtics
  set xlabel 'E - E_F (eV)'
  set label '{/Symbol G}' at g1,ymin-offset center
  set label 'K' at k,ymin-offset center
  set label 'M' at m,ymin-offset center
  set label '{/Symbol G}' at g2,ymin-offset center
  set xrange [xmin:xmax]
  set yrange [ymin:ymax]
  set xzeroaxis lt 0
  plot 'gr.bands.gnu' using (\$1):(\$2-ef)
  pause -1

  and type

  sh band_plot.sh

  Here the output file for the corresponding SCF calculation is assumed to be "gr.scf.out."