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

Question

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.

Answer 1

Answer: band folding is a little more subtle than I originally thought. I'd only really come across band folding in the textbook example of 1D and rather naively assumed that in higher dimensions if you were 'ignoring' the dimension that had a supercell that it could be neglected. This is emphatically not the case.

An example that illustrates this: Consider a tight binding model in 2D, where we have a supercell along x of length 2. The hoppings in all directions are, say, 0.1 and the onsite energies 1. We neglect the hoppings along y for now; the (folded) band structure along k_x is plainly given by:

Now we turn on the hoppings along y. We will be interested in the bandstructure along the (0, k_y) line. My naive assertion was that as k_x = 0 we ignore the supercell folding, but this is incorrect. The above graph has 2 energies at the Gamma point, as under the folding k_x = pi is identified with k_x = 0. So, the complete bandstructure should look something like this:

And if we take the desired cut along (0, k_y) we see essentially the same thing I was seeing with my example above:

This of course solves the problem. Many thanks for the help! The minimal answer is I guess that folding in general changes the number of bands at the gamma point, so even if you're setting that direction to zero it still contributes.