  We propose a new type of Josephson junction formed by two superconductors
close to the superconductor-Mott-insulator transition, one of which is doped
with holes and the other is doped with electrons. A self-organized
Mott-insulating depletion region is formed at the interface between two
superconductors, giving rise to an asymmetric response of current to the
external voltage. The collective excitations of the depletion region result in
a novel phase dynamics that can be measured experimentally in the noise
spectrum of the Josephson current.
