Temperature-driven sodium-ion dynamical-to-static crossover in the zig-zag ordered phase of Na$_{0.5}$CoO$_2$

We employ polarization-resolved Raman spectroscopy combined with first-principles calculations to study the sodium-ion lattice dynamics in a sodium zig-zag ordered cobaltate compound Na$_{0.5}$CoO$_2$. We detect two sodium phonon modes for the first time, and their mode frequencies are consistent with first-principles phonon calculations based on an orthorhombic unit cell. We find that they appear below around $T^*\sim300\pm50$K with large linewidth broadening, much lower than the sodium zig-zag ordering temperature $T_\text{S}\sim460$K, and then narrow at lower temperatures. We interpret the sodium-phonon anomalies occurring at $T^*$ as a dynamical-to-static crossover involving mainly the motion of sodium ions. Our results suggest that the gradual freezing of the sodium ions and the well-defined static sodium-zigzag order below $T^*$ set the stage for the emergent electronic and magnetic orders in the CoO$_2$ layer of Na$_{0.5}$CoO$_2$.