In this letter, a bipolar resistive-switching random-access memory (RRAM) in Ni/Si3N4/SiO2/p+-Si structure and its fabrication process are demonstrated. The proposed device with double-layer dielectrics consisting of Si3N4 layer (5 nm) as a resistive switching and SiO2 (2.5 nm) layer for the tunnel barrier is investigated in comparison with that having a single layer of Si3N4. Double-layer cell shows ultra-low power operation under a compliance current (ICOMP) of 500 nA, which ensures the reset current (IRESET) of sub-1 μA much lower than that of the single-layer cell. Also, large on/off ratio (∼105) has been obtained since the SiO2 layer efficiently suppresses the current in the high-resistance state. Moreover, maximum selectivity in double-layer cell is 122 when 1/2 read bias scheme is applied to the crossbar array. Highly nonlinear I-V characteristics of the double-layer Si3N4-based RRAM cell warrant the realization of selector-free RRAM cell in the crossbar array pursuing higher integration density.