# Brillouin zones of a compound after structural relaxation

My background in solid-state chemistry isn't much, so I apologize in advance if there arises the need to raise a basic fact while answering my question. I've been learning mostly from the web, with my main text resource 'Band Theory of Solids: An Introduction from the point of view of symmetry.' by Simon L. Altmann.

From what I understand, Brillouin zones are areas (or volumes in 3-D) in reciprocal space where you use to describe the energy levels with respect to the k-vectors. Thus typically in band structure plots the k-vectors range from the origin Γ along some straight path which contains points like Δ, K etc.

So far the common examples I've seen for a Brillouin zone are for 'pure' elements (Si, Cu etc.) where you have the atoms in a crystal formation, which you obtain direct lattice pattern, and then the reciprocal space, and finally the Brillouin zone polyhedron.

What I'm really curious about is, if I have a material that I know the Brillouin zone (like a graphite sheet), and this material reacts with another molecule (like a dopant), how would the Brillouin zone change? A scenario would be performing structural relaxation of a carbon nanotube with some dopants, and the nanotube deforms slightly (with the dopants moving to an equilibrium configuration). In this case even if I know the Brillouin zone of a pure (undoped) carbon nanotube, with the deformation, how should I determine which are the k-vector points then?

• The answer is neither general nor easy. First, doped structures are non-periodic, therefore you need an assumption how you treat it. Second, the dopant will introduce new energy levels, and interact with the existing ones. How will they look like? You often have to make calculations to have a clue, no trivial answer for that. – Greg Aug 25 '17 at 0:53
• If the doped structures are periodic (say I arrange them in a periodic configuration and the relaxed model turns out to be periodic as well), would that be still possible to do that? I was able to obtain relaxation calculations of the doped structures, but to have the software (I use SIESTA) obtain the band structure values I would need to input the band lines or band points, which is what I'm stuck at, since I don't exactly know if the Brillouin zone is the same (as the carbon nanotube), and if the X, K points still match.. – Khai Yi Aug 25 '17 at 17:10