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. The symmetry points are given in the unit of basic reciprocal lattice vectors.
  • Post-processing
    To plot the band structure with gnuplot or alike, "bands.x" is used. Here's an example of the input file (pp_bands.in):
    &bands
     outdir  = './tmp'
     prefix  = 'gr'
     filband = 'gr.bands'
     lsym    = .true.
    /
    Note that the same number of processors as the previous band structure calculation should be used. 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 $5}'`
    gnuplot <<EOF
    ef=${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 ylabel '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
    EOF
    and type
    sh band_plot.sh
    Here the output file for the corresponding SCF calculation is assumed to be "gr.scf.out." If you want to save the band structure change the terminal in gnuplot as
    #!/bin/sh
    ef=`grep Fermi gr.scf.out | awk '{print $5}'`
    gnuplot <<EOF
    ef=${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 ylabel '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
    set terminal postscript eps
    set output 'gr.bands.eps'
    plot 'gr.bands.gnu' using (\$1):(\$2-ef)
    pause -1
    EOF
    The figure for the band structure is written to 'gr.bands.eps.'
    gr.bands.png

Exercise

The position of the Dirac cone at the K-point should be located at the Fermi level. However, the position of the Fermi level is very sensitive to the number of k-point. Examine how many k-points (how dense the k-point mesh) starting from, for e.g., 6x6x1, 8x8x1, ... 12x12x1.

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Last-modified: 2019-11-28 (木) 12:22:59 (12d)