How to calculate Band Structure of Silicon with QE alone

Silicon's band structure, computed with Quantum ESPRESSO alone -- a type 1, independent calculation with the QELK suite (no Elk involved here; for the direct Elk-vs-QE overlay, see the Si_band_elk_vs_qe example).

 

The input file

 

We need only the input file

Si_band_qe.qelk

version 1

cell bohr
  -5.1   0.0   5.1
   0.0   5.1   5.1
  -5.1   5.1   0.0
end

atoms crystal
  Si  0.00  0.00  0.00
  Si  0.25  0.25  0.25
end

species Si
  elk_file   Si.in
  qe_pseudo  Si.pz-vbc.UPF
  qe_mass    28.086
end

step scf qe
  from initial
  calculation  scf
  ecutwfc      12.0
  kpoints      2 2 2
end

step bands qe
  from initial
  calculation  bands
  ecutwfc      20.0
  nbnd         8
  kpath
    0.5 0.5 0.5 1  L
    0.4 0.4 0.4 2
    0.3 0.3 0.3 3
    0.2 0.2 0.2 4
    0.1 0.1 0.1 5
    0.0 0.0 0.0 6  G
    0.0 0.0 0.1 7
    0.0 0.0 0.2 8
    0.0 0.0 0.3 9
    0.0 0.0 0.4 10
    0.0 0.0 0.5 11
    0.0 0.0 0.6 12
    0.0 0.0 0.7 13
    0.0 0.0 0.8 14
    0.0 0.0 0.9 15
    0.0 0.0 1.0 16  X
    0.0 0.1 1.0 17
    0.0 0.2 1.0 18
    0.0 0.3 1.0 19
    0.0 0.4 1.0 20
    0.0 0.5 1.0 21  W
    0.0 0.4 0.8 22
    0.0 0.3 0.6 23
    0.0 0.2 0.4 24
    0.0 0.1 0.2 25
    0.0 0.0 0.0 26  G
  end
end

Note: two steps, not one -- unlike Elk's single-step task 0 20, QE's bands step needs a separate, prior scf step: bands reads the converged charge density QE itself wrote to outdir/prefix.save. Both steps share the suite's default outdir/prefix, so bands picks it up with no extra wiring. ecutwfc differs on purpose (12.0 for the cheap scf density, 20.0 for the bands diagonalisation) -- QE allows a higher wavefunction cutoff for a non-self-consistent bands run, since only the diagonalisation basis changes, not the density being read in.

 

Calculation

 

Si_band_qe$ qelk Si_band_qe.qelk --dry-run
[scf] (qe) started from cell=[-5.1, 0.0, 5.1]... , 2 atoms
[bands] (qe) started from cell=[-5.1, 0.0, 5.1]... , 2 atoms

All step directories are under: run/

algerien1970@linux-ml0a:~/abinitio/QELK/examples/Si_band_qe> qelk Si_band_qe.qelk

 

Plot

 

plot_qe_bands.py bands.pw.out --qelk Si_band_qe.qelk --step bands -o qe_bands.png

Note: unlike Elk, QE's band energies sit on QE's own absolute eV scale, not referenced to the Fermi level -- that is why the plot below runs from about -5.6 eV to +16.6 eV rather than being centred on 0.

 

SCF result

 

grep -E "convergence has been achieved|^!|Fermi energy" scf.pw.out

     the Fermi energy is     6.8058 ev
!    total energy              =     -15.61421023 Ry
     convergence has been achieved in   5 iterations

 

Analysis

 

Eight bands, plotted along L - G - X - W - G. At G itself (bands.pw.out, k = 0.0000 0.0000 0.0000):

-5.6071   6.5350   6.5350   6.5350   8.9678   8.9678   8.9678   9.9992   (eV)

One low, non-degenerate valence band (-5.61 eV), then a THREEFOLD degenerate valence band (6.535 eV, all three exactly equal), then a threefold degenerate conduction band (8.968 eV), then a single higher band (9.999 eV). That fourfold split -- 1, 3, 3, 1 -- is exactly the group-theory signature of the diamond structure's T_d site symmetry at G, and it is the same 1-low-band + 3-degenerate-band valence structure the Elk-only tutorial found independently, on a completely different basis set and a completely different absolute energy scale. That agreement between two unrelated numerical methods is a much stronger check than either code's number alone.

Four bands sit below the gap at every k-point with no partial occupation -- 2 atoms x 4 valence electrons (3s²3p²) = 8 electrons = 4 doubly-occupied bands, confirming a semiconductor, not a metal.

Scanning all 26 k-points directly in bands.pw.out for the valence-band maximum (highest 4th band) and conduction-band minimum (lowest 5th band):

VBM = 6.535 eV, at G       (band 4, exact -- next closest point is 6.18 eV)
CBM = 6.847 eV, at (0,0,0.6), between G and X on the G-X line
indirect gap = 0.312 eV

The valence maximum and conduction minimum sit at different k-points -- G for the top of the valence bands, roughly 60% of the way from G to X for the bottom of the conduction bands -- which is exactly what makes silicon an indirect-gap semiconductor, and matches the real Si conduction-band valleys sitting along G-X (the textbook value is nearer 85% of the way to X; this coarse 0.1-spaced k-path only samples every tenth of the line, so it lands close to, not exactly on, the true minimum).

Note: 0.312 eV is in the right ballpark for plain LDA silicon (typically quoted around 0.5-0.6 eV, against an experimental gap of 1.17 eV -- LDA is well known to systematically underestimate semiconductor gaps). The Fermi energy from the scf step, 6.8058 eV, sits just above this bands run's CBM (6.847 eV) rather than in the middle of the gap -- a plausible consequence of the scf step's very coarse 2x2x2 k-mesh (chosen for a fast density, not a converged Fermi level) rather than anything wrong with the bands calculation itself.

Above the gap, the bands broaden out quickly -- values above 8-9 eV climb past 16 eV by the middle of the G-X segment -- consistent with QE's plane-wave conduction states becoming increasingly free-electron-like away from the zone centre, the same qualitative picture textbook silicon band structures show.