Tetrahedral amorphous (ta-C) carbon-based memory devices have recently gained traction due to their good scalability and promising properties like nanosecond switching speeds. However, cycling endurance is still a key challenge. In this paper, we present a model that takes local fluctuations in sp2 and sp3 content into account when describing the conductivity of ta-C memory devices. We present a detailed study of the conductivity of ta-C memory devices ranging from ohmic behavior at low electric fields to dielectric breakdown. The study consists of pulsed switching experiments and device-scale simulations, which allows us for the first time to provide insights into the local temperature distribution at the onset of memory switching.