Memristors are attracting considerable interest for their prospective applications in nonvolatile memory, neuromorphic computing, and in-memory computing. However, the nature of resistance switching is still under debate, and current fluctuation in memristors is one of the critical concerns for stable performance. In this work, random telegraph noise (RTN) as the indication of current instabilities in distinct resistance states of the Pt/Ti/HfO2/W memristor is thoroughly investigated. Standard two-level digital-like RTN, multilevel current instabilities with non-correlation/correlation defects, and irreversible current transitions are observed and analyzed. The dependence of RTN on the resistance and read bias reveals that the current fluctuation depends strongly on the morphology and evolution of the conductive filament composed of oxygen vacancies. Our results link the current fluctuation behaviors to the evolution of the conductive filament and will guide continuous optimization of memristive devices.

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