In this work, we investigated the gradual switching and self-rectifying characteristics of Cu/alpha-IGZO/p(+)-Si resistive-switching random-access memory (RRAM) device. We fabricated the RRAM cells with Cu as the top electrode (TE) and heavily doped p-type silicon as the bottom electrode (BE), and amorphous indium gallium zinc oxide (alpha-IGZO) film as the switching layer. In particular, we developed a bilayer IGZO film consisting of an oxygen-deficient layer and an oxygen-rich one by controlling the oxygen concentrations in the respective switching layers in the expectation of gradual switching owing to an oxygen vacancy reservoir. Fabricated RRAM cells successfully showed the typical hysteretic I-V curves including SET and RESET operations in the DC sweep mode. Furthermore, gradual switching and self-rectifying performances were observed. These characteristics are suitable to applications for synaptic devices toward the advanced neuromorphic systems.