Gallium arsenide's band structure, computed with Quantum ESPRESSO alone -- a type 1, independent calculation with the QELK suite (no Elk involved here; the Elk-only counterpart is the GaAs_band_elk example).
The input file
We need only the input file
GaAs_band_qe.qelk
version 1
cell bohr
-5.3400 0.0000 5.3400
0.0000 5.3400 5.3400
-5.3400 5.3400 0.0000
end
atoms crystal
Ga 0.00 0.00 0.00
As 0.25 0.25 0.25
end
species Ga
elk_file Ga.in
qe_pseudo Ga.pbe-dn-kjpaw_psl.0.2.upf
qe_mass 69.723
end
species As
elk_file As.in
qe_pseudo As.pbe-n-kjpaw_psl.0.2.upf
qe_mass 74.922
end
step scf qe
from initial
calculation scf
ecutwfc 40
kpoints 12 12 12
nbnd 13
occupations tetrahedra_opt
end
step bands qe
from scf
calculation bands
ecutwfc 40
nbnd 16
kpath
0.35356 0.35356 0.35356 1 L
0.28284 0.28284 0.28284 2
0.21213 0.21213 0.21213 3
0.14142 0.14142 0.14142 4
0.07071 0.07071 0.07071 5
0.00000 0.00000 0.00000 6 G
0.00000 0.00000 0.07071 7
0.00000 0.00000 0.14142 8
0.00000 0.00000 0.21213 9
0.00000 0.00000 0.28284 10
0.00000 0.00000 0.35356 11
0.00000 0.00000 0.42427 12
0.00000 0.00000 0.49498 13
0.00000 0.00000 0.56569 14
0.00000 0.00000 0.63640 15
0.00000 0.00000 0.70711 16 X
0.00000 0.07071 0.70711 17
0.00000 0.14142 0.70711 18
0.00000 0.21213 0.70711 19
0.00000 0.28284 0.70711 20
0.00000 0.35356 0.70711 21 W
0.00000 0.42427 0.70711 22
0.00000 0.49498 0.70711 23
0.00000 0.56569 0.70711 24
0.00000 0.63640 0.70711 25
0.00000 0.70711 0.70711 26
0.00000 0.63640 0.63640 27
0.00000 0.56569 0.56569 28
0.00000 0.49498 0.49498 29
0.00000 0.42427 0.42427 30
0.00000 0.35356 0.35356 31
0.00000 0.28284 0.28284 32
0.00000 0.21213 0.21213 33
0.00000 0.14142 0.14142 34
0.00000 0.07071 0.07071 35
0.00000 0.00000 0.00000 36 G
end
end
Note: two steps: QE's bands step needs a prior scf step, and declares it with from scf -- a bands step may follow only a QE scf (or nscf) step. bands reads the scf step's charge density through the shared default outdir/prefix. Both steps use the same ecutwfc (40 Ry) and PBE pseudopotentials; nbnd 16 in the bands step adds empty conduction bands to the plot.
Note: QE's kpath is Cartesian in units of 2Ï€/alat, and alat is the LENGTH OF THE FIRST CELL VECTOR (7.5519 bohr here, printed in scf.pw.out as "lattice parameter (alat)"), not the cubic constant 10.68 bohr. The familiar L = (.5 .5 .5), X = (0 0 1), W = (0 .5 1) are therefore multiplied by 0.70711 (= alat/a), which gives the numbers above (L = 0.35356 0.35356 0.35356, X = 0 0 0.70711, W = 0 0.35356 0.70711). The same path is used by the Elk-only example, written in Elk's own lattice coordinates.
Calculation
algerien1970@linux-ml0a:~/abinitio/QELK/examples/GaAs_band_qe> qelk GaAs_band_qe.qelk
A first check with --dry-run writes both pw.in files without launching any binary:
algerien1970@linux-ml0a:~/abinitio/QELK/examples/GaAs_band_qe> qelk GaAs_band_qe.qelk --dry-run
SCF result
grep -E "convergence has been achieved|^!|Fermi energy|number of electrons" scf.pw.out
number of electrons = 18.00
! total energy = -453.96442749 Ry
the Fermi energy is 7.9132 ev
convergence has been achieved in 8 iterations
PBE (SLA PW PBX PBC), 72 irreducible k-points on the 12x12x12 mesh (tetrahedron method), ecutwfc 40 Ry and a charge-density cutoff of 320 Ry (qelk writes ecutrho = 8 x ecutwfc, the usual choice for PAW). Eighteen electrons -- Ga 13 and As 5 in these pseudopotentials -- fill nine bands. Unlike Elk, the As pseudopotential keeps no 3d shell in the valence.
Plot
plot_qe_bands.py bands.pw.out --qelk GaAs_band_qe.qelk --step bands -o qe_bands.png
Note: unlike Elk, QE's band energies are on QE's own absolute eV scale, not relative to the Fermi level -- the plot below runs from about -7 eV to +22 eV. To put the Fermi energy at 0 eV, pass it explicitly: --efermi $(grep "Fermi energy" scf.pw.out | awk '{print $5}').
Analysis
Sixteen bands along L - G - X - W - G. At G (bands.pw.out, k = 0 0 0), in eV:
-7.182 -7.182 -7.182 -7.103 -7.103 -5.116 7.626 7.626 7.626 8.147 11.347 11.347 11.347 15.306 17.787 17.787
bands 1-5 : -7.18 (x3) and -7.10 (x2) -- the Ga 3d shell, split 3 + 2 by the tetrahedral field band 6 : -5.12 -- the lowest valence band, mostly As 4s bands 7-9 : +7.626, threefold -- the p-derived valence bands (valence top) band 10 : +8.147, single -- the lowest conduction band bands 11-13: +11.347, threefold
Nine occupied bands, as the 18 electrons require. The three valence-top bands are exactly degenerate at G, then split along L and X; the conduction minimum is a single level, the same pattern as in the Elk-only tutorial.
valence band maximum : 7.626 eV, at G conduction band min : 8.147 eV, at G direct gap at G : 0.521 eV
Both extrema sit at G: GaAs is a direct-gap semiconductor, unlike silicon. The Fermi energy from the scf step, 7.9132 eV, lies inside the gap (0.287 eV above the valence maximum, 0.234 eV below the conduction minimum). The gap, 0.521 eV, is the usual PBE value, well below the experimental 1.42 eV.
Comparing with the Elk-only tutorial, which used the same cell, the same path and the same functional:
Elk QE direct gap at G (eV) 0.520 0.521 valence bandwidth, 4s band to VBM 12.74 12.74 (QE: 7.626 - (-5.116)) Ga 3d below the valence maximum -14.8 to -14.6 -14.8 to -14.7 (eV, Elk: bands 6-10) conduction level above the gap at G 3.446 - 0.245 = 3.20 11.347 - 8.147 = 3.20
The two all-electron and pseudopotential descriptions agree to about 0.01 eV on the gap and the bandwidth, and to about 0.1 eV on the position of the Ga 3d level -- the same physical picture computed in two very different ways. Only the As 3d shell (near -35 eV in Elk) has no counterpart in QE, because the As pseudopotential freezes it.
Note: QE's curves show small kinks because QE was asked for only 36 k-points along the path (against Elk's 200).
My name is Dr Abderrahmane Reggad. I'm a 54 year old and I am a researcher in materials' sciences. 
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