Charge and Spin Ordering in Insulator Na$_{0.5}$CoO$_2$: Effects of Correlation and Symmetry

The discovery by Takada and coworkers of superconductivity in Na$_{0.3}$CoO$_2 \bullet$1.3 H$_2$O near 5K has led to extensive studies of the rich variation of properties in the Na$_x$CoO$_2$ system ($0.2 \le x \le 1$), which has a triangular lattice of Co sites and a layered structure. In addition, specifically at $x$=0.5, the system has been observed to undergo a charge disproportionation (2Co$^{3.5+} \rightarrow$ Co$^{3+}$+Co$^{4+}$) and metal-insulator transition at 50 K, while the rest of the phase diagram is metallic. We will present results of studies of charge disproportionation and charge- and spin-ordering in insulating in Na$_{0.5}$CoO$_2$, applying ab initio band theory including correlations due to intra-atomic repulsion. Various ordering patterns (zigzag and two striped) for four-Co supercells are analyzed before focusing on the observed ``out-of-phase stripe'' pattern of antiferromagnetic Co$^{4+}$ spins along charge-ordered stripes. This pattern relieves frustration and shows distinct analogies with the cuprate layers: a bipartite lattice of antialigned spins, with axes at 90$^{\circ}$ angles. Substantial distinctions with cuprates are also discussed, including the tiny gap of a new variant of ``charge transfer'' type within the Co $3d$ system. [References] [1] K. Takada et al., Nature {\bf 422}, 53 (2003). [2] M. L. Foo et al., Phys. Rev. Lett. {\bf 92},247001 (2004). [3] K.-W. Lee, J. Kunes, P. Novak, and W. E. Pickett, Phys. Rev. Lett. {\bf 94}, 026403 (2005). [4] K.-W. Lee and W. E. Pickett, cond-mat/0510555.