We demonstrate the potential of a graphene capacitor structure on silicon-rich nitride micro-ring resonators for multitasking operations within high performance computing. Capacitor structures formed by two graphene sheets separated by a 10 nm insulating silicon nitride layer are considered. Hybrid integrated photonic structures are then designed to exploit the electro-absorptive operation of the graphene capacitor to tuneably control the transmission and attenuation of different wavelengths of light. By tuning the capacitor length, a shift in the resonant wavelength is produced giving rise to a broadband multilevel photonic volatile memory. The advantages of using silicon-rich nitride as the waveguiding material in place of the more conventional silicon nitride (Si3N4) are shown, with a doubling of the device's operational bandwidth from 31.2 to 62.41 GHz achieved while also allowing a smaller device footprint. A systematic evaluation of the device's performance and energy consumption is presented. A difference in the extinction ratio between the ON and OFF states of 16.5 dB and energy consumptions of <0.3 pJ/bit are obtained. Finally, it has been demonstrated that increasing the permittivity of the insulator layer in the capacitor structure, the energy consumption per bit can be reduced even further. Overall, the resonance tuning enabled by the novel graphene capacitor makes it a key component for future multilevel photonic memories and optical routing in high performance computing.