Quantum dots have discrete electronic band structures (see https://www.worldscientific.com/doi/pdf/10.1142/9789814397988_0004#:~:text=In%20the%20quantum%20dot%2C%20there,4.1) due to having very few interacting atoms, this leads me to think thin films should have discrete electronic band structures 'out of plane', is this reasonable?
Also, expanding on this, would nano wires have discrete band structures ‘out of line’, with continuous band structures ‘in line’
I am not an expert in quantum dots, but here are some hints. The quantum dots have an atomic behavior, a band structure is generated by the translational operator $\hat{T}$ whatever the dimension of the dimension $D>0$. Obviously is 0$D$, this translational symmetry is broken, you cannot built a Bloch state and the result are discrete atomic-like levels, their number and spacing depends mainly on the number of atoms. For any other dimension, there is a translational symmetry, no matter the shape of the Brillouin zone, there is a Bloch state and therefore a band structure for thin films, monolayers (but there is no band structure "out of plane") or nanowires.
