Magnetic-field-induced Fermi surface reconstruction in Na$_{0.5}$CoO$_2$

We have performed an electrical transport study in Na$_{0.5} $CoO$_{2}$ at high fields $B$ and low temperatures $T$. We find that the charge ordered state observed below $T_{CO}$ = 53 K can be suppressed by large in- plane magnetic fields, but not by fields applied along the inter-plane direction. For $B$ rotating within the conductive CoO$_{2}$ layers we observe angular magnetoresistance oscillations of essentially two-fold periodicity consistent with the reported orthorhombic symmetry of Na$_{0.5} $CoO$_{2}$. As $B$ increases however, a new 6-fold periodicity emerges indicating the stabilization of a hexagonal FS as reported by the ARPES measurements. This observation suggests on the one hand, that the Na superstructure defines the geometry of the FS at low temperatures, and on the other, that the charge order in the conducting plane is suppressed by high in-plane fields. At low temperatures Shubnikov de Haas oscillations (SdH) of very small frequencies are observed for $B$ // c-axis, indicating that almost the entire FS reported for $x$ = 0.6 and 0.7 disappears below $T_{CO}$ for $x$ = 0.5. Our results strongly indicate that the charge ordering involves the coupling with the Na order and involves the large hole pocket.