in the direction of the $X_{\rm{SC}}$ point in the simple cubic BZ, despite both $X$ points being in the centre of a face of their BZ. Due to the rotation, what was the $\Delta_{\rm{SC}}$ direction in the simple cubic BZ is now the $\Sigma_{\rm{TET}}$ direction (pointing towards $M$, at the corner of the BZ in the $k_{z}=0$ plane) in the tetragonal BZ. The tetragonal CBM, while physically still the same as the CBM in the FCC or simple cubic BZ, is not represented in the same fashion (see Fig. \[fig:d0vasp\]).

![Geometrical difference between the simple cubic and tetragonal cells; $\left(0 0 1\right)$ planar cut through an atomic monolayer.[]{data-label="fig:rotation"}](rotation){width="0.5\linewidth"}

![The Brillouin zone for a tetragonal cell. The $M$–$\Gamma$–$X$ path used in this work is shown.[]{data-label="fig:tetBZ"}](brillouin){width="0.6\linewidth"}

![Band structure diagram for the tetragonal bulk Si structures with increasing number of layers, calculated using the <span style="font-variant:small-caps;">vasp</span> plane wave method (see Sec. \[sec:methods\]).[]{data-label="fig:d0vasp"}](d0vasp){width="\linewidth"}

Band folding in $z$-direction {#sec:zfold}
=============================

  -------------------------------------------------------- -------- ------------ ------------- --------------------------
                           Basis                            No. of     No. of        LUMO                 CBM
                            type                            layers    *k*-pts     at $\Gamma$   (at $\Delta_{\rm{FCC}}$)
                                                                     in $k_{z}$      (eV)                 (eV)
                             PW                               4          12         0.7517     
    (<span style="font-variant:small-caps;">vasp</span>)      8          6          0.7517     
                                                              16         3          0.6506     
                                                              32         2          0.6170     
                                                              40         1          0.6179     
                                                              64         1          0.6137     
                                                              80         