Phase noise analysis and control of VO$_2$-based relaxation type oscillators

VO$_2$-based relaxation oscillators form a rapidly developing field that finds applications in neuromorphic computing, Ising machines, and numerous signal processing concepts. These oscillators operate in a deeply nonlinear relaxation regime based on rapid phase transitions between insulating and metallic states in the VO$_2$ material. This process is governed by thermal effects, which lead to additional voltage fluctuations and contribute to a considerably wide spectral linewidth in the VO$_2$-based oscillator signal. In this work, we thoroughly study the phase noise in VO$_2$-based relaxation oscillators and demonstrate that the broadening of the generation spectrum linewidth at low oscillation frequencies is caused by an increased susceptibility to thermal fluctuations during the incubation phase. We explore the types of noise affecting oscillator stability and show that synchronization with an external square-wave signal improves the phase noise more effectively than a sinusoidal-shape injection locking signal.