How big does the Schottky Barrier or Band Offset have to be for a junction to be a diode?
The answer is "anything"; however, that does not mean that it is useful as a diode.  All of the "ohmic" contacts on a semiconductor are actually very thin eV barriers.  If I put a metal plug down to polysilicon, I get a Schottky contact, but then I keep increasing doping until the barrier is thin enough to look like a resistor due to the small energy barrier potential.