  The function of nano-scale devices critically depends on the choice of
materials. For electron transport junctions it is natural to characterize the
materials by their conductance length dependence, $\beta$. Theoretical
estimations of $\beta$ are made employing two primary theories: complex band
structure and DFT-NEGF Landauer transport. Both reveal information on $\beta$
of individual states; i.e. complex Bloch waves and transmission eigenchannels,
respectively. However, it is unclear how the $\beta$-values of the two
approaches compare. Here, we present calculations of decay constants for the
two most conductive states as determined by complex band structure and standard
DFT-NEGF transport calculations for two molecular and one semi-conductor
junctions. Despite the different nature of the two methods, we find strong
agreement of the calculated decay constants for the molecular junctions while
the semi-conductor junction shows some discrepancies. The results presented
here provide a template for studying the intrinsic, channel resolved length
dependence of the junction through complex band structure of the central
material in the heterogeneous nano-scale junction.
