  We present a comprehensive first-principles study of the electronic charge
redistribution in atomically sharp LaAlO$_3$/SrTiO$_3$(001) heterointerfaces of
both n- and p-types allowing for non-stoichiometric composition. Using two
different computational methods within the framework of the density functional
theory (linear combination of atomic orbitals and plane waves) we demonstrate
that conducting properties of LaAlO$_3$/SrTiO$_3$(001) heterointerfaces
strongly depend on termination of LaAlO$_3$ (001) surface. We argue that both
the polar "catastrophe" and the polar distortion scenarios may be realized
depending on the interface stoichiometry. Our calculations predict that
heterointerfaces with a non-stoichiometrc film---either LaO-terminated n-type
or AlO$_2$-terminated p-type---may exhibit the conductivity of n- or p-type,
respectively, independently of LaAlO$_3$(001) film thickness.
