# Different band structure in supercell, but not due to band folding

This is my first time asking a question - if I mess something up please let me know and I'll amend. I believe this goes beyond questions on how to use QE so here seems appropriate over the QE forums.

I've been running a lot of DFT simulations with QuantumESPRESSO, and run into an interesting issue with supercells. To be clear, I am not talking about band folding (unless I'm very mistaken). I'm looking at surface relaxations and surface band structures, and have found that a supercell band structure comes out different even when the kpath does not include any components in the direction of the increase in size of the unit cell.

I've attached a quick example with Sodium, comparing the unit cell and 1x1x2 supercell. This is not using relaxed values in the name of convenience, but the error persists in my main investigations with fully relaxed/converged values. The supercell is 1x1x2 along z, and the k-path (slightly peculiarly, I know) does not include anything in the kz direction. As you can see, there are twice as many bands and they're all over the place.

I know that supercells slightly break QE's symmetry detection algorithm, so it is presumably guessing the irrep content wrong. However, physically we should expect these extra bands to be 'accidentally' degenerate with the bulk bands of the conventional unit cell.

Inputs:

Unit cell scf

&CONTROL
calculation = 'scf'
restart_mode = 'from_scratch'
prefix = 'Na'
outdir = './outdir/'
pseudo_dir = '/pseudodir/'
etot_conv_thr = 1e-8
/
&SYSTEM
ibrav=1, celldm(1) = 8.111,
nat = 2, ntyp = 1,
nbnd = 25
ecutrho = 325,
ecutwfc = 80,
occupations = 'smearing'
smearing = 'gaussian'
degauss = 0.01
/
&ELECTRONS
conv_thr=1e-8
!mixing_beta = 0.5
/
ATOMIC_SPECIES
Na     22.99   Na.pbe-spn-rrkjus_psl.1.0.0.UPF

ATOMIC_POSITIONS (alat)
Na 0.000000 0.000000 0.000000
Na 0.5 0.5 0.5

K_POINTS (automatic)
12 12 12 0 0 0


Unit cell bands

&CONTROL
calculation = 'bands'
restart_mode = 'from_scratch'
prefix = 'Na'
outdir = './outdir/'
pseudo_dir = '/pseudodir/'
etot_conv_thr = 1e-8
/
&SYSTEM
ibrav=1, celldm(1) = 8.111,
nat = 2, ntyp = 1,
nbnd = 25,
ecutrho = 325,
ecutwfc = 80,
occupations = 'smearing'
smearing = 'gaussian'
degauss = 0.01
/
&ELECTRONS
conv_thr=1e-8
mixing_beta = 0.5
/
ATOMIC_SPECIES
Na     22.99   Na.pbe-spn-rrkjus_psl.1.0.0.UPF

ATOMIC_POSITIONS (alat)
Na 0.000000 0.000000 0.000000
Na 0.5 0.5 0.5

!weird k path with no z
K_POINTS {crystal_b}
5
0 0 0 20 !G
0.5 -0.5 0 20 !H
0 0.5 0 20 !N
0.25 0.25 0 20 !P
0 0 0 1 !G


Supercell scf

&CONTROL
calculation = 'scf'
restart_mode = 'from_scratch'
prefix = 'NaSC'
outdir = './outdir/'
pseudo_dir = '/pseudodir/'
etot_conv_thr = 1e-8
/
&SYSTEM
ibrav=6, celldm(1) = 8.111, celldm(3) = 2,
nat = 4, ntyp = 1,
nbnd = 50
ecutrho = 325,
ecutwfc = 80,
occupations = 'smearing'
smearing = 'gaussian'
degauss = 0.01
/
&ELECTRONS
conv_thr=1e-8
!mixing_beta = 0.5
/
ATOMIC_SPECIES
Na     22.99   Na.pbe-spn-rrkjus_psl.1.0.0.UPF

ATOMIC_POSITIONS (alat)
Na 0.000000 0.000000 0.000000
Na 0.5 0.5 0.5
Na 0 0 1
Na 0.5 0.5 1.5
K_POINTS (automatic)
12 12 6 0 0 0


Supercell bands

&CONTROL
calculation = 'bands'
restart_mode = 'from_scratch'
prefix = 'NaSC'
outdir = './outdir/'
pseudo_dir = '/pseudodir/'
etot_conv_thr = 1e-8
/
&SYSTEM
ibrav=6, celldm(1) = 8.111, celldm(3) = 2,
nat = 4, ntyp = 1,
nbnd = 50,
ecutrho = 325,
ecutwfc = 66,
occupations = 'smearing'
smearing = 'gaussian'
degauss = 0.01
/
&ELECTRONS
conv_thr=1e-8
mixing_beta = 0.5
/
ATOMIC_SPECIES
Na     22.99   Na.pbe-spn-rrkjus_psl.1.0.0.UPF

ATOMIC_POSITIONS (alat)
Na 0.000000 0.000000 0.000000
Na 0.5 0.5 0.5
Na 0 0 1
Na 0.5 0.5 1.5
K_POINTS {crystal_b}
5
0 0 0 20 !G
0.5 -0.5 0 20 !H
0 0.5 0 20 !N
0.25 0.25 0 20 !P
0 0 0 1 !G


And the band structures:

Unit cell: Supercell:

Has anyone seen this before? Can anyone help explain why this happens? Even better, can anyone fix this? Thanks.

Edit 1: Thank you to Camps for pointing out an error - I believe I've fixed this now.

• Welcome to our community! Mar 23, 2022 at 16:51
• Why do you think this is not band folding? If you double the size of the cell in any direction, you expect to get twice the number of states you had in the original cell. Mar 26, 2022 at 20:53
• You know what... you're right. So: my thoughts were that yes there should indeed be twice as many states, but that they should be degenerate with the already existing bands. So there'd be the same bands but with twice the degeneracy. I was writing something to explain why I thought I was right and realised that the way band folding works is a little more subtle than I'd assumed and this is actually incorrect. Honestly thank you! I can write an answer as I think it's useful, I'd like to give you the tick though if possible. Mar 28, 2022 at 10:07
• A minor comment: it's often helpful to plot the points without joining them up, because plotting programs have no idea which points join to which, but the lines can fool you into thinking bands cross when they don't, or vice versa. In your example, if you ignore the lines then you can see the supercell and original cell's bands match perfectly, but the plotting program has mis-joined some supercell bands near crossing points. Aug 5, 2022 at 23:00
• How did things go? Have you found an answer now? Perhaps you could write a self-answer with what you've learned about this problem over the last 5.5 months? Were the suggestions by Phil helpful? It would be nice to get this out of the unanswered queue. Please update us! Sep 7, 2022 at 14:04