We demonstrated fully vertical Schottky barrier diodes (SBDs) that have a Si-doped AlN drift layer directly grown on n-type 4H-SiC substrate by metal-organic chemical-vapor deposition. The AlN SBD with a Ni anode showed a clear rectifying characteristic at 300-500 K and rectification ratio of about 102. We found that the leakage current of the vertical AlN-on-SiC devices is affected by defects in the AlN drift layer and Schottky interface.