outside the new BZ boundary (past $X^{'}$); but a valid solution in any BZ must be a solution in all BZs. This results in the phenomenon of band folding, whereby a band continuing past a BZ boundary reenters the BZ on the opposite side. Since the $X$ direction in a face-centred cubic (FCC) BZ is 6-fold symmetric, a solution near the opposite BZ boundary is also a solution near the one we are focussing on. This results in the appearance that the band continuing past the BZ boundary is “reflected”, or folded, back on itself into the first BZ. Since the new BZ boundary in this direction is now at $k_{\rm{BZ}}^{'}=X^{'}=0.5\frac{2\pi}{a}$, the location of the valley will be at $k_{0}^{'}=X^{'}-\left(k_{0}-X^{'}\right)\sim0.15\frac{2\pi}{a}$, as mentioned in Ref. . Each further increase in the size of the supercell will result in more folding (and a denser bandstructure). Care is therefore required to distinguish between a new band and one which has been folded due to this effect when interpreting bandstructure.

Continuing with our example of silicon, whilst the classic bandstructure [@Chelikowsky74] is derived from the bulk Si primitive FCC cell (containing two atoms), it is often more convenient to use a simple cubic (SC) 