  Monitoring electronic properties of 2D materials is an essential step to open
a way for applications such as electronic devices and sensors. From this
perspective, Bernal bilayer graphene (BLG) is a fairly simple system that
offers great possibilities for tuning electronic gap and charge carriers'
mobility by selective functionalization (adsorptions of atoms or molecules).
Here, we present a detailed numerical study of BLG electronic properties when
two types of adsorption site are present simultaneously. We focus on realistic
cases that could be realized experimentally with adsorbate concentration c
varying from 0.25% to 5%. For a given value of c, when the electronic doping is
lower than c we show that quantum effects, which are ignored in usual
semi-classical calculations, strongly affect the electronic structure and the
transport properties. A wide range of behaviors is indeed found, such as gap
opening, metallic behavior or abnormal conductivity, which depend on the
adsorbate positions, the c value, the doping, and eventually the coupling
between midgap states which can create a midgap band. These behaviors are
understood by simple arguments based on the fact that BLG lattice is bipartite.
We also analyze the conductivity at low temperature, where multiple scattering
effects cannot be ignored. Moreover, when the Fermi energy lies in the band of
midgap states, the average velocity of charge carriers cancels but conduction
is still possible thanks to quantum fluctuations of the velocity.
