  PrMnO$_3$ (PMO) based RRAM shows selector-less behavior due to
high-non-linearity. Recently, the non-linearity, along with volatile
hysteresis, is demonstrated and utilized as a compact oscillator to enable
highly scaled oscillatory neurons, which enable oscillatory neuromorphic
systems found in the human cortex. Hence, it is vital to understand the
physical mechanisms behind such a volatile hysteretic behavior to provide
useful insights in developing a device for various neuromorphic applications.
In this paper, we present a comprehensive investigation of the transient
characteristics and propose a physical mechanism to replicate the observations
by simulations. First, we investigate the complex dynamics of the hysteresis
with the voltage ramp rate. We observe that the voltage window initially
increases and later decreases as the ramp rate is increased - while the current
window reduces monotonically. Second, an analytical electrothermal model based
on space charge limited current (SCLC) and Fourier Heat equation is proposed to
model the co-dependent heat and current flow. Finally, we show that the
interplay between the self-heating due to the current and the current
dependence on the temperature is accurately modeled to reproduce the hysteresis
dependence on the various voltage ramp rate. Such a detailed understanding of
the device PMO RRAM volatile hysteresis may enable efficient device design
required in neuromorphic computing applications.
