  We report first principles modeling of quantum tunneling through amorphous
HfO$_2$ dielectric layer of metal-oxide-semiconductor (MOS) nanostructures in
the form of n-Si/HfO$_2$/Al. In particular we predict that chemically modifying
the amorphous HfO$_2$ barrier by doping N and Al atoms in the middle region -
far from the two interfaces of the MOS structure, can reduce the
gate-to-channel tunnel leakage by more than one order of magnitude. Several
other types of modification are found to enhance tunneling or induce
substantial band bending in the Si, both are not desired from leakage point of
view. By analyzing transmission coefficients and projected density of states,
the microscopic physics of electron traversing the tunnel barrier with or
without impurity atoms in the high-$\kappa$ dielectric is revealed.
