the final density of states available to the scattered particle increases with the particle energy (for non-parabolic energy band dispersions in 2D), this increase is not enough to offset the reduction in the capture rates due to the factors mentioned above.

Since the energy width of the valence and conduction bands in MoS$_{2}$ are less than 1.2 eV and 0.6 eV [@Lam12; @Louie1], respectively, the limited horizontal extents of the curves in Fig.\[fig:capture1\] ensure that the electron (hole) scattered to a high energy in the hole (electron) capture process is scattered within the same band consistent with the assumptions made in this work. It is, however, possible for the scattered particle to go into a different band. For example, slightly away from the $K$ ($K'$) points, the next higher conduction band has Bloch states with a large weight on the $d_{z^{2}}$ orbitals of Mo atoms and these Bloch states will have large overlap with the Bloch states near the conduction band bottom [@Guinea13]. It should also be noted that the weights $|b_{c}|^{2}$ and $|b_{v}|^{2}$ for defects could be very small or zero. For example, in the case of sulfur vacancy $A_{1}$ states both $|b_{c}|^{2}$ and $|b_{v}|^{2}$ are expected to be 