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I am performing some DFT calculations of monolayer hexagonal boron nitride in Quantum ESPRESSO. I have been trying to calculate the density of states of a 7 x 7 supercell of h-BN, but the resulting density of states does not scale up properly. For a 1 x 1 primitive cell system, I get a smooth plot; for a 3 x 3 system, I get an identical plot, but for a 7 x7 system, there is too much noise:

enter image description here enter image description here enter image description here

What might be the cause of this trouble? The only difference between the scf and nscf files for the supercells is the number of K-POINTS and total number of atoms.

scf file for 1 x 1:

&CONTROL
  calculation = 'scf'
  prefix = 'hbn'
  pseudo_dir = '../pseudos/'
  outdir = '../out/'
/

&SYSTEM
  ibrav = 0
  nat = 2
  ntyp = 2
  occupations = 'smearing'
  smearing = 'mv'
  degauss =   0.002
  ecutwfc =   60
  ecutrho =   600
/

&ELECTRONS
  mixing_beta =   0.7
  conv_thr =      1d-6
/
ATOMIC_SPECIES
    B      10.811       B.pbe-n-kjpaw_psl.1.0.0.UPF
    N      14.0067      N.pbe-n-kjpaw_psl.1.0.0.UPF

ATOMIC_POSITIONS crystal
    B            0.6666666666       0.3333333333       0.5000000000 
    N            0.3333333333       0.6666666666       0.5000000000 

CELL_PARAMETERS (angstrom)
   2.514856081   0.000000000   0.000000000
  -1.257428041   2.177929253   0.000000000
   0.000000000   0.000000000  10.000000000

K_POINTS automatic
  9 9 1 0 0 0

scf for 3 x 3 case:

&CONTROL
  calculation = 'scf'
  prefix = 'hbn'
  pseudo_dir = './pseudos/'
  outdir = './out/'
/

&SYSTEM
  ibrav = 0
  nat = 18
  ntyp = 2
  occupations = 'smearing'
  smearing = 'mv'
  degauss =   0.002
  ecutwfc =   60
  ecutrho =   600
/

&ELECTRONS
  mixing_beta =   0.7
  conv_thr =      1d-6
/
ATOMIC_SPECIES
    5      10.811       B.pbe-n-kjpaw_psl.1.0.0.UPF
    7      14.0067      N.pbe-n-kjpaw_psl.1.0.0.UPF

ATOMIC_POSITIONS angstrom
    5    1.2574280400    0.7259764180    5.0000000000
    7    0.0000000000    1.4519528350    5.0000000000
    5   -0.0000000010    2.9039056710    5.0000000000
    7   -1.2574280410    3.6298820880    5.0000000000
    5   -1.2574280420    5.0818349240    5.0000000000
    7   -2.5148560820    5.8078113410    5.0000000000
    5    3.7722841210    0.7259764180    5.0000000000
    7    2.5148560810    1.4519528350    5.0000000000
    5    2.5148560800    2.9039056710    5.0000000000
    7    1.2574280400    3.6298820880    5.0000000000
    5    1.2574280390    5.0818349240    5.0000000000
    7   -0.0000000010    5.8078113410    5.0000000000
    5    6.2871402020    0.7259764180    5.0000000000
    7    5.0297121620    1.4519528350    5.0000000000
    5    5.0297121610    2.9039056710    5.0000000000
    7    3.7722841210    3.6298820880    5.0000000000
    5    3.7722841200    5.0818349240    5.0000000000
    7    2.5148560800    5.8078113410    5.0000000000 

CELL_PARAMETERS (angstrom)
   7.544568243   0.000000000   0.000000000
  -3.772284123   6.533787759   0.000000000
   0.000000000   0.000000000  10.000000000

K_POINTS automatic
  3 3 1 0 0 0

scf file for 7 x 7 case:

&CONTROL
  calculation = 'scf'
  prefix = 'hbn'
  pseudo_dir = './pseudos/'
  outdir = './out/'
/

&SYSTEM
  ibrav = 0
  nat = 98
  ntyp = 2
  occupations = 'smearing'
  smearing = 'mv'
  degauss =   0.002
  ecutwfc =   60
  ecutrho =   600
/

&ELECTRONS
  mixing_beta =   0.7
  conv_thr =      1d-6
/
ATOMIC_SPECIES
    5      10.811       B.pbe-n-kjpaw_psl.1.0.0.UPF
    7      14.0067      N.pbe-n-kjpaw_psl.1.0.0.UPF

ATOMIC_POSITIONS angstrom
    5    1.2574280400    0.7259764180    5.0000000000
    7    0.0000000000    1.4519528350    5.0000000000
    5   -0.0000000010    2.9039056710    5.0000000000
    7   -1.2574280410    3.6298820880    5.0000000000
    5   -1.2574280420    5.0818349240    5.0000000000
    7   -2.5148560820    5.8078113410    5.0000000000
    5   -2.5148560830    7.2597641770    5.0000000000
    7   -3.7722841230    7.9857405940    5.0000000000
    5   -3.7722841240    9.4376934300    5.0000000000
    7   -5.0297121640   10.1636698470    5.0000000000
    5   -5.0297121650   11.6156226830    5.0000000000
    7   -6.2871402050   12.3415991000    5.0000000000
    5   -6.2871402060   13.7935519360    5.0000000000
    7   -7.5445682460   14.5195283530    5.0000000000
    5    3.7722841210    0.7259764180    5.0000000000
    7    2.5148560810    1.4519528350    5.0000000000
    5    2.5148560800    2.9039056710    5.0000000000
    7    1.2574280400    3.6298820880    5.0000000000
    5    1.2574280390    5.0818349240    5.0000000000
    7   -0.0000000010    5.8078113410    5.0000000000
    5   -0.0000000020    7.2597641770    5.0000000000
    7   -1.2574280420    7.9857405940    5.0000000000
    5   -1.2574280430    9.4376934300    5.0000000000
    7   -2.5148560830   10.1636698470    5.0000000000
    5   -2.5148560840   11.6156226830    5.0000000000
    7   -3.7722841240   12.3415991000    5.0000000000
    5   -3.7722841250   13.7935519360    5.0000000000
    7   -5.0297121650   14.5195283530    5.0000000000
    5    6.2871402020    0.7259764180    5.0000000000
    7    5.0297121620    1.4519528350    5.0000000000
    5    5.0297121610    2.9039056710    5.0000000000
    7    3.7722841210    3.6298820880    5.0000000000
    5    3.7722841200    5.0818349240    5.0000000000
    7    2.5148560800    5.8078113410    5.0000000000
    5    2.5148560790    7.2597641770    5.0000000000
    7    1.2574280390    7.9857405940    5.0000000000
    5    1.2574280380    9.4376934300    5.0000000000
    7   -0.0000000020   10.1636698470    5.0000000000
    5   -0.0000000030   11.6156226830    5.0000000000
    7   -1.2574280430   12.3415991000    5.0000000000
    5   -1.2574280440   13.7935519360    5.0000000000
    7   -2.5148560840   14.5195283530    5.0000000000
    5    8.8019962830    0.7259764180    5.0000000000
    7    7.5445682430    1.4519528350    5.0000000000
    5    7.5445682420    2.9039056710    5.0000000000
    7    6.2871402020    3.6298820880    5.0000000000
    5    6.2871402010    5.0818349240    5.0000000000
    7    5.0297121610    5.8078113410    5.0000000000
    5    5.0297121600    7.2597641770    5.0000000000
    7    3.7722841200    7.9857405940    5.0000000000
    5    3.7722841190    9.4376934300    5.0000000000
    7    2.5148560790   10.1636698470    5.0000000000
    5    2.5148560780   11.6156226830    5.0000000000
    7    1.2574280380   12.3415991000    5.0000000000
    5    1.2574280370   13.7935519360    5.0000000000
    7   -0.0000000030   14.5195283530    5.0000000000
    5   11.3168523640    0.7259764180    5.0000000000
    7   10.0594243240    1.4519528350    5.0000000000
    5   10.0594243230    2.9039056710    5.0000000000
    7    8.8019962830    3.6298820880    5.0000000000
    5    8.8019962820    5.0818349240    5.0000000000
    7    7.5445682420    5.8078113410    5.0000000000
    5    7.5445682410    7.2597641770    5.0000000000
    7    6.2871402010    7.9857405940    5.0000000000
    5    6.2871402000    9.4376934300    5.0000000000
    7    5.0297121600   10.1636698470    5.0000000000
    5    5.0297121590   11.6156226830    5.0000000000
    7    3.7722841190   12.3415991000    5.0000000000
    5    3.7722841180   13.7935519360    5.0000000000
    7    2.5148560780   14.5195283530    5.0000000000
    5   13.8317084450    0.7259764180    5.0000000000
    7   12.5742804050    1.4519528350    5.0000000000
    5   12.5742804040    2.9039056710    5.0000000000
    7   11.3168523640    3.6298820880    5.0000000000
    5   11.3168523630    5.0818349240    5.0000000000
    7   10.0594243230    5.8078113410    5.0000000000
    5   10.0594243220    7.2597641770    5.0000000000
    7    8.8019962820    7.9857405940    5.0000000000
    5    8.8019962810    9.4376934300    5.0000000000
    7    7.5445682410   10.1636698470    5.0000000000
    5    7.5445682400   11.6156226830    5.0000000000
    7    6.2871402000   12.3415991000    5.0000000000
    5    6.2871401990   13.7935519360    5.0000000000
    7    5.0297121590   14.5195283530    5.0000000000
    5   16.3465645260    0.7259764180    5.0000000000
    7   15.0891364860    1.4519528350    5.0000000000
    5   15.0891364850    2.9039056710    5.0000000000
    7   13.8317084450    3.6298820880    5.0000000000
    5   13.8317084440    5.0818349240    5.0000000000
    7   12.5742804040    5.8078113410    5.0000000000
    5   12.5742804030    7.2597641770    5.0000000000
    7   11.3168523630    7.9857405940    5.0000000000
    5   11.3168523620    9.4376934300    5.0000000000
    7   10.0594243220   10.1636698470    5.0000000000
    5   10.0594243210   11.6156226830    5.0000000000
    7    8.8019962810   12.3415991000    5.0000000000
    5    8.8019962800   13.7935519360    5.0000000000
    7    7.5445682400   14.5195283530    5.0000000000 

CELL_PARAMETERS (angstrom)
   17.603992567  0.000000000   0.000000000
  -8.801996287   15.245504771  0.000000000
   0.000000000   0.000000000  10.000000000

K_POINTS automatic
  3 3 1 0 0 0
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1 Answer 1

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When the size of the supercell increases, you can reduce the number of k-points in an inverse manner to get the same accuracy as the one with the smaller cell. This is because the relation between real space and k-space is reciprocal.

For a unit (i.e. $1\times1\times1$) cell, you used $9\times9\times1$ k-grid. So for a $3\times3\times1$ supercell, your new k-point should be $3\times3\times1$ which you correctly used. However, for a $7\times7\times1$ supercell, you still used the same k-grid of $3\times3\times1$ and this increased the k-point density than you had in your previous calculations. That's a probable cause of the noise you see in your DOS.

Moreover, it is not clear if you used the same k-grid in your NSCF calculation or if you increased it a little bit with occupations='tetrahedra'. The k-grid of NSCF can decide the DOS output significantly. So my suggestions would be to fix the k-point scaling, and if there is nothing special with a $7\times7\times1$ supercell, then use a supercell where you can calculate the scaling to be an integer. And also check the following suggested k-points distance information from MaterialsCloud's QE input generator. I think for a little bit larger supercell, a gamma-only or Baldereschi point-only 1/4 1/4 1/4 (crystal) sampling will suffice in your case.

The default is 0.2 1/Å, that is a conservative, all-purpose choice. We call this "fine" sampling, and the appropriate smearing/degauss is 0.2 eV. The two other options are "very fine" sampling (0.15 1/Å, 0.1 eV), or "normal" (0.3 1/Å, 0.3 eV). Note that we change the degauss since a coarser sampling benefits from more smearing. Last: there is no point in sampling the Brillouin Zone in a direction in which there shouldn't be any dispersion (as is the case for, say, a 0-, 1- or 2-dimensional system where 3, 2, or 1 directions shouldn't be sampled); in these cases you should update the input file by hand a posteriori. You can also decide to use a shifted Monkhorst-Pack mesh (just change 0 0 0 -> 1 1 1 in the last 3 numbers of the k-point entry). If you are a PWscf Jedi, for large cells you could use the Baldereschi point (1/4 1/4 1/4) and nosym=.true. to get much better sampling than Gamma-only (at twice the cost).

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  • $\begingroup$ Thanks for the insightful answer! For the NSCF calculations, I used a K-POINT grid which is 4 times as dense (e.g. 12 12 1, if for scf it is 3 3 1). I have also used tetrahedra occupations in NSCF. How would I ensure that my k-point scaling is fixed? It is not easy to divide 9 by other integers (apart from 3). $\endgroup$ Mar 15 at 13:17
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    $\begingroup$ About the scaling, you can definitely use a gamma-point only (1 1 1) sampling for a 9 9 1 supercell. You can try a gamma point only, or a 2 2 1 sampling in your 7 7 1 supercell maybe but I am not fully sure it would give you identical results but pretty sure it would be better. But for NSCF, 4 times dense k-mesh looks like an overkill. Sometimes when the supercell is large enough (more than 10 Angstrom or so), I don't even change the k-grid in NSCF at all. It still gives me very good DOS output. $\endgroup$ Mar 15 at 18:24
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    $\begingroup$ Thanks Abdul! I will try this out and let you know of the result. $\endgroup$ Mar 16 at 3:19
  • $\begingroup$ So it turns out, that I could fix my problem by simply including a small smearing amount in the dos.x post-processing step for all my files, to get the DOS output identical for all the three cases. $\endgroup$ Apr 1 at 1:11

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