Band structure calculations are typically carried out in a non-self-consistent manner, so you can't in the standard sense with whatever code you're using. I.e. your code does not allow for an scf calculation followed by an nscf calculation with a fixed density from the scf calculation, but on a different set of k-points (at least for hybrid functionals).

This basically means you'd have to do the calculation for the full bandpath self consistently.

For the band structure calculation in the hybrid potential, first, the band structure calculation should be done geometrically from the hybrid potential. Band structure is available through the analysis option in Castep. Then, if the file is exported to Excel, the graph can be drawn at the origin with that data.

Yes, quantum espresso cannot run a non-self consistent calculation on hybrid functional. You have two options then:

1) You give to QE a path instead of a set of k-points for the scf calculation. This however might produce inaccurate results as you are not sampling the Brillouin zone properly. What you can also do is superimpose a regular mesh with the points in the path. Then extract these points and their relative energies from the scf calculation.

2) You use a regular mesh, but increase its size. Then utilise an interpolation code (for example Boltztrap2 or Ypp from the Yambo project) to extract the band structure along the path of your interest. The clear disadvantage here is that you need to sample the BZ with a fine mesh making the calculation with the hybrid functional heavier than normal.

Finally, you might take a ”common” approach: Calculate with the hybrid functional the gap, but use the dispersion from the a standard functional (the one from which the hybrid is derived) when you open the band gap by hand. This however might not work since the relative position of the maxima and minima might depend on the hybrid functional.

I hope this helps,

Roberto