Low-Frequency Noise and Resistive Switching in $\beta$-Na${}_{0.33}$V${}_2$O${}_5$

The interplay between charge ordering and its manifestation in macroscopic electrical transport in low-dimensional materials is crucial for understanding resistive switching mechanisms. In this study, we investigate the electronic transport and switching behavior of single-crystalline $\beta$-Na${}_{0.33}$V${}_2$O${}_5$, focusing on low-frequency resistance noise dynamics of charge-order-driven resistive switching. Using electrical transport, low frequency noise spectroscopy, and X-ray diffraction, we probe electron dynamics across the Na-ion-ordering (IO) and charge-ordering (CO) transitions. Near room temperature, the weak temperature dependence of the noise spectral density points to a dominance of nearest-neighbor polaron hopping. Below IO transition temperature ($T_{IO}\sim 240\text{K}$), structural analysis reveals that Na-ions adopt a zig-zag occupancy pattern, breaking the two-fold rotational symmetry and influencing the electronic ground state. Subsequently, a sharp drop in resistance noise below the CO transition temperature ($T_{CO}\sim 125\text{K}$) indicates the emergence of correlated electron behavior. Furthermore, application of sufficient electric field leads to the destabilization of the CO state, and a transition to a high-conducting state. The material exhibits distinct resistive switching between 35~K and 110~K, with a resistance change spanning two orders of magnitude, primarily driven by electronic mechanisms rather than Joule heating. These findings provide new insights into charge-order-induced switching and electronic correlations in quasi-one-dimensional systems, with potential applications in cryogenic memory and neuromorphic computing devices owing to the low noise levels in their stable resistive states.