bands (see figure \[fig:dirac\](a) (inset)) extends along $k_z$ over the whole Brillouin zone as the derived Bi $p$ bands are quasi-2D. This is further supported by partial conductivity calculations where we have integrated over cylindrical regions in $k$-space centered around points shown in the inset in Fig. \[fig:dirac\](a). The radius of each cylinder was chosen to be $\approx 0.1\times\frac{2\pi}{a}$ and the results clearly indicate that the major contribution to this peak is from the D-point which is between the “lens” and the “boomerang” shaped Fermi surfaces where the Fermi level lies in the gap. Calculations also indicate that the magnitude of this gap does not depend on $k_z$ as expected. The contribution from B and X-points are also quite significant but as these points are in close proximity to the D-point, it is not surprising because the cylindrical volumes are not mutually exclusive. This is the reason why these spectra are not additive.

![(a) Decomposition of the the inter-band contribution to the optical conductivity from the transition between the bands (37,38) $\rightarrow$ (39,40) from different parts of the Fermi surface (fig. inset) (b) Linear fit of the inter-band transition peaked at 950cm$^{-1}$.[]{data-label="fig:dirac"}](dirac_dec.png "fig:"){width="48.00000%"}\
![(a) Decomposition of the the inter-band contribution 