Amorphous InGaZnO (a-IGZO) is promising for transparent electronics due to its high carrier mobility and optical transparency. However, most metal/a-IGZO junctions are ohmic due to the Fermi-level pinning at the interface, restricting their device applications. Here, we report that indium-tin oxide/a-IGZO Schottky diodes can be formed by gradient oxygen doping in the a-IGZO layer that would otherwise form an ohmic contact. Making use of back-to-back a-IGZO Schottky junctions, a transparent IGZO permeable metal-base transistor is also demonstrated with a high common-base gain.