Magnetic-Field-Induced Suppression of the Charge Ordered State in Na$_{0.5}$CoO$_{2}$ and the observation of Shubnikov--de Haas Oscillations in Na$_{x}$CoO$_{2}$

We performed electrical transport measurements at low temperatures and high magnetic fields in Na$_{x}$CoO$_{2}$ single crystals for both $x$ = 0.5 and $x$ = 0.3. For $x$ = 0.5 Shubnikov de Haas oscillations corresponding to only 1{\%} of the area of the orthorhombic Brillouin zone (BZ) were clearly observed, indicating that most of the original Fermi surface vanishes at the charge ordering (CO) transition. While in-plane magnetic fields were found to strongly suppress the charge ordered state observed for x = 0.5 via a field-induced strongly hysteretic transition. When the external fields are rotated within the conducting planes, we observe angular magnetoresistance oscillations whose periodicity changes from two-to six-fold at the transition suggesting the reconstruction of the Fermi surface of this material. These facts indicate that the charge order is a delicate one, more akin to a charge-density-wave, and consistent with the small gap observed in the optical conductivity. While for $x$ = 0.3 we clearly observe quantum oscillatory phenomena for two frequencies $f_{1} \cong $ 480 and $f_{2}$ $\cong $ 800 T corresponding respectively to only 0.8 and 1.36{\%} of the first Brillouin zone (FBZ), with very weak indications of possible additional frequencies. These values contrast markedly with what is predicted by band structure calculations for $x$ = 0.3, i.e., 2.26{\%} and 22.3{\%} of the FBZ for the pockets resulting from the $e_{g}$$'$ and the $a_{1g}$ bands, respectively. We speculate that the Na superstructures seen for both concentrationsm re-define the Brillouin zone and thus the geometry of the Fermi surface explaining perhaps such discrepancies.