How does a Schottky diode work?
First it depends on the type of semiconductor(SC) n and p types are different.  I'll just do n-type.  (p-type is the mirror image of n-type).  Then, in theory, it only depends on the relative work function of the metal and the SC.  If the metal is below the SC then electrons can flow and no diode.  But if the metal is above the SC then you get a depletion region in the SC, and a diode.  

Oh, in practice other things, like surface states play a role.  (At least that's my limited understanding.)